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Indicator function

The indicator function, also known as the characteristic function in some mathematical contexts, of a subset A of a set X is a function \mathbf{1}_A: X \to \{0, 1\} (or more generally to \mathbb{R}) that maps every element x \in A to 1 and every element x \notin A to 0.^[1]^[2] This binary-valued function serves as a foundational tool in various branches of mathematics, providing a precise way to encode membership in a set.^[3] In set theory and combinatorics, indicator functions enable elegant computations of set cardinalities and intersections; for instance, the size of a set A \subseteq U is given by |A| = \sum_{x \in U} \mathbf{1}_A(x), and the product rule \mathbf{1}_A \cdot \mathbf{1}_B = \mathbf{1}_{A \cap B} underpins the inclusion-exclusion principle for unions of sets.^[1] They are particularly useful in deriving formulas like Bonferroni inequalities and Jordan's identity for counting elements across multiple sets.^[1] In probability theory and measure theory, the indicator function \mathbf{1}_A of an event A in a probability space becomes a random variable, often denoted I_A, with expectation \mathbb{E}[I_A] = P(A), linking probabilities directly to integrals via Lebesgue measure.^[3]^[2] Properties such as \mathbf{1}_{A \cup B} = \mathbf{1}_A + \mathbf{1}_B - \mathbf{1}_A \cdot \mathbf{1}_B and \mathbf{1}_{A^c} = 1 - \mathbf{1}_A facilitate variance calculations, moment-generating functions, and approximations like the central limit theorem for sums of independent indicators, which model binomial distributions.^[3] These applications extend to advanced topics, including Fourier analysis of indicator functions for convex sets and optimization problems involving \ell_1-norm minimization reformulated via indicator constraints.^[4]^[5]

Fundamentals

Definition

In mathematics, the indicator function of a subset A of a universal set X is a function $1_A: X \to \{0,1\} defined by

$1_A(x) = \begin{cases} 1 & \text{if } x \in A, \\ 0 & \text{if } x \notin A. \end{cases}

^[6] This binary-valued mapping directly encodes set membership, serving as the simplest non-constant function that distinguishes elements of A from those outside it, thereby providing a foundational tool for representing subsets in functional terms.^[7] The concept emerged in the 19th century, with an early explicit use by Peter Gustav Lejeune Dirichlet in his 1829 paper on the convergence of trigonometric series, where he introduced the Dirichlet function as the indicator of the rational numbers within the reals to illustrate discontinuities in Fourier representations.^[8] It gained further prominence in early 20th-century integration theory, where it formed the basis for defining measurable functions and simple functions in the Lebesgue sense.^[7] The indicator function represents the canonical binary step function, taking a constant value of 1 on A and 0 elsewhere, in contrast to more general step functions, which are finite linear combinations of such indicators over disjoint intervals.

Notation and Terminology

The indicator function of a subset A of a set X, which takes the value 1 if x \in A and 0 otherwise, is denoted in various ways across mathematical literature. In set theory, the notation \chi_A(x) is standard, reflecting its role as the characteristic function that distinguishes membership in A.^[9] This usage dates back to early 20th-century discussions of set-theoretic functions, such as the characteristic function of the rationals in analyses of Baire classes.^[10] In contrast, modern analysis and measure theory often employ $1_A(x) or I_A(x), emphasizing the function's binary output in integration and measurability contexts.^[11] A compact alternative notation, the Iverson bracket [P], assigns 1 to a proposition P if true and 0 if false, generalizing the indicator for logical conditions.^[12] For instance, the indicator for even integers can be written as [n \equiv 0 \pmod{2}], facilitating succinct expressions in sums and identities.^[13] Terminologically, "indicator function" predominates in contemporary analysis, while "characteristic function" prevails in set theory, especially in pre-1950s texts where it described membership without modern probabilistic connotations.^[14] In measure theory, $1_A is a field-specific preference for clarity in integral definitions, as seen in standard references.^[11] Researchers should note potential confusion with the unrelated "characteristic function" in probability theory, defined as \phi(t) = \mathbb{E}[e^{itX}] for a random variable X.^[14]

Properties

Basic Properties

The indicator function $1_A of a set A in a universe X evaluates pointwise to 1 if the argument x belongs to A and to 0 otherwise, taking values exclusively in the set \{[0](/page/0), [1](/page/1)\}.^[15] This binary nature reflects membership status directly. Furthermore, $1_A(x) = 1 - 1_{X \setminus A}(x) for all x \in X, linking the indicator to its complement set.^[15] The support of $1_A, defined as the set where the function is nonzero, coincides with A, while it vanishes exactly on the complement X \setminus A.^[16] This localization property underscores the function's role in identifying set boundaries without additional structure.^[16] Logical operations on sets translate to arithmetic operations on their indicators. Specifically, for arbitrary sets A and B,

$1_{A \cap B}(x) = 1_A(x) \cdot 1_B(x)

for all x, since the product yields 1 only when both factors are 1.^[15] Similarly,

$1_{A \cup B}(x) = 1_A(x) + 1_B(x) - 1_A(x) \cdot 1_B(x),

which equals 1 if at least one input is in the respective set, accounting for overlap via subtraction of the intersection term; equivalently, it is \max(1_A(x), 1_B(x)).^[15] These equivalences hold pointwise and mirror Boolean logic for membership. The notation $1_A is standard, though \chi_A is sometimes used interchangeably.^[15] The indicator function exhibits idempotence under pointwise multiplication: (1_A(x))^2 = 1_A(x) for all x, as squaring preserves the values 0 and 1.^[15] This algebraic property aligns with the idempotence of set intersection, A \cap A = A.^[15]

Arithmetic and Set Operations

The indicator function of the intersection of two sets satisfies $1_{A \cap B} = 1_A \cdot 1_B pointwise, as the product equals 1 only when both indicators are 1, i.e., when the point lies in both sets.^[17] This multiplicative property holds for finite or infinite intersections, reflecting the logical AND operation in arithmetic form.^[18] For disjoint sets A_i, the indicator of their union is the sum of the indicators: $1_{\cup A_i} = \sum 1_{A_i}, since the sets do not overlap and each point belongs to at most one A_i.^[19] This additivity extends to finite or countable disjoint unions and underpins the counting measure, where the measure of a set A is \mu(A) = \int 1_A \, d\mu = \sum_{x \in A} 1, equating integration to summation over points.^[20] The inclusion-exclusion principle for the indicator of a union, $1_{\cup A_i} = \sum (-1)^{k+1} \sum 1_{\cap_{j=1}^k A_{i_j}} over intersections of k sets, arises as a special case of Möbius inversion on the power set lattice, where the Möbius function \mu(S, T) = (-1)^{|T \setminus S|} for S \subseteq T inverts the zeta function to yield the exact union indicator.^[21]^[22] Indicator functions form a basis for the vector space of simple functions in L^p spaces ($1 \leq p < \infty), as any simple function is a finite linear combination \sum c_j 1_{E_j} of indicators of measurable sets E_j with finite measure, spanning the dense subspace of step functions used in integration and approximation.^[23]

Applications in Probability and Statistics

Expectation, Variance, and Covariance

In probability theory, the indicator function $1_A for an event A in a probability space serves as a simple random variable that takes the value 1 if A occurs and 0 otherwise. Its expectation is precisely the probability of the event: \mathbb{E}[1_A] = P(A). This follows directly from the definition of expectation as \mathbb{E}[1_A] = 1 \cdot P(A) + 0 \cdot (1 - P(A)) = P(A).^[24]^[25] The variance of $1_A can be derived using the general formula \mathrm{Var}(1_A) = \mathbb{E}[1_A^2] - (\mathbb{E}[1_A])^2. Since $1_A^2 = 1_A (as the indicator takes values in {0,1}), it holds that \mathbb{E}[1_A^2] = \mathbb{E}[1_A] = P(A), yielding \mathrm{Var}(1_A) = P(A) - [P(A)]^2 = P(A)(1 - P(A)). This expression highlights the indicator's Bernoulli-like behavior, with maximum variance at P(A) = 1/2.^[26]^[27] For two events A and B, the covariance between their indicators is \mathrm{Cov}(1_A, 1_B) = \mathbb{E}[1_A 1_B] - \mathbb{E}[1_A] \mathbb{E}[1_B]. Note that $1_A 1_B = 1_{A \cap B}, so \mathbb{E}[1_A 1_B] = P(A \cap B), and thus \mathrm{Cov}(1_A, 1_B) = P(A \cap B) - P(A)P(B). This measures the dependence between events: if A and B are independent, then P(A \cap B) = P(A)P(B) and \mathrm{Cov}(1_A, 1_B) = 0, implying zero correlation \rho(1_A, 1_B) = 0. Conversely, if A and B are mutually exclusive (P(A \cap B) = 0), the covariance is negative, -P(A)P(B), leading to negative correlation, as seen in problems like matching cards where fixed points exhibit repulsion.^[28]^[25] A key property is the linearity of expectation, which states that for any collection of events A_1, \dots, A_n (possibly dependent), \mathbb{E}\left[\sum_{i=1}^n 1_{A_i}\right] = \sum_{i=1}^n \mathbb{E}[1_{A_i}] = \sum_{i=1}^n P(A_i). This holds without requiring independence and is particularly useful for approximating inclusion-exclusion probabilities, such as bounding the probability of unions via the expected number of occurrences. For disjoint events, this aligns with basic additivity, where \sum 1_{A_i} = 1_{\cup A_i}.^[24]^[29]

Role in Indicator Random Variables

In probability theory, an indicator function serves as the foundation for defining indicator random variables, which model the occurrence of events in a probability space. Given a probability space (\Omega, \mathcal{F}, P) and an event A \in \mathcal{F}, the indicator random variable I_A: \Omega \to \{0, 1\} is defined by I_A(\omega) = 1 if \omega \in A and I_A(\omega) = 0 otherwise, thereby capturing whether the event A occurs for a given sample point \omega.^[30] This construction allows indicator random variables to represent binary outcomes in stochastic models, such as success or failure in trials.^[31] The distribution of an indicator random variable I_A is Bernoulli with parameter p = P(A), denoted I_A \sim \operatorname{Bern}(p). The probability mass function is given by

P(I_A = 1) = p, \quad P(I_A = 0) = 1 - p,

reflecting the probability of the event occurring or not.^[31] This Bernoulli structure underscores the role of indicator functions in modeling rare or binary events, where the expected value E[I_A] = p directly equals the event probability.^[30] Indicator random variables are central to the Poisson paradigm, which approximates the distribution of the sum of many rare, weakly dependent indicators by a Poisson distribution. If X = \sum_{i=1}^n I_{A_i} where the events A_i have small probabilities p_i = P(A_i) and limited dependence, then X \approx \operatorname{Poisson}(\mu) with \mu = \sum p_i = E[X], providing a useful approximation for counting rare events like defects or mutations.^[32] This paradigm extends the classical binomial-to-Poisson limit by handling dependence through bounds on total variation distance.^[33] Stein's method enhances these approximations by quantifying the error in Poisson approximations for sums of indicators, even under dependence. The Stein-Chen approach provides explicit bounds on the total variation distance, enabling precise error control in applications like reliability analysis or network traffic modeling.^[34] In stochastic processes, sums of indicator random variables define counting processes that track event occurrences over time. For a renewal process, the counting process N(t) counts the number of renewals up to time t and can be expressed as N(t) = \sum_{n=1}^\infty I_{\{S_n \leq t\}}, where S_n = X_1 + \cdots + X_n are the partial sums of independent, positive interarrival times X_i.^[35] This representation facilitates analysis in renewal theory, such as deriving the renewal function m(t) = E[N(t)] = \sum_{n=1}^\infty P(S_n \leq t), which quantifies long-run behavior like the expected number of system repairs.^[36]

Specialized Uses in Mathematics

In Recursion Theory and Logic

In the 1930s, amid David Hilbert's program to formalize mathematics and prove the consistency of axiomatic systems using finitary methods, foundational work in recursion theory by Kurt Gödel, Alonzo Church, Stephen Kleene, and Alan Turing revealed profound limits on computability and provability.^[37] This era's developments, including Gödel's incompleteness theorems (1931) and Turing's analysis of the halting problem (1936), demonstrated undecidability results that relied on precise encodings and decision predicates, where indicator functions served as binary classifiers for computational and logical properties.^[37] These contributions shifted focus from Hilbert's optimism toward understanding the boundaries of effective procedures in logic. In recursion theory, characteristic functions play a central role in defining recursive sets and linking to undecidability. A set A \subseteq \mathbb{N} is recursive if and only if its characteristic function \chi_A, defined by \chi_A(x) = 1 if x \in A and $0 otherwise, is a total recursive function.^[38] For the diagonal halting set K = \{ e \mid \phi_e(e) \downarrow \}, where \phi_e is the e-th partial recursive function and \downarrow denotes convergence (halting), the characteristic function \chi_K is not recursive, as proven by Turing's diagonalization argument showing no algorithm can decide membership in K for all e.^[38] This non-recursiveness directly establishes the undecidability of the halting problem, highlighting how indicator functions capture the binary nature of termination but fail to be computable in general cases.^[38] Gödel's β-function further illustrates the use of indicator-like structures in logical encodings. Defined as \beta(c, a, i) = \text{rem}(c, 1 + a(i+1)), where \text{rem} is the remainder function and a is a sequence of natural numbers greater than 1, the β-function encodes finite sequences \langle n_0, n_1, \dots, n_k \rangle into a single natural number c via the Chinese Remainder Theorem, ensuring unique decodability.^[39] In Gödel's incompleteness proofs, this encoding arithmetizes syntax, enabling the definition of the provability predicate \operatorname{Prov}(y), which acts as an indicator: \operatorname{Prov}(y) holds (true) if y is the Gödel number of a provable formula in the system, and false otherwise, formalized as \exists x \, \operatorname{Prf}(x, y) where \operatorname{Prf}(x, y) indicates x codes a proof of y.^[40] Such predicates were crucial for showing that no consistent formal system can prove all truths about arithmetic, tying indicator mechanisms to undecidability.^[41] Kleene's T-predicate extends this framework to partial recursive functions, providing a primitive recursive relation that flags definedness. The Normal Form Theorem states there exists a primitive recursive predicate T(e, \mathbf{x}, y) such that \phi_e(\mathbf{x}) \downarrow = z if and only if T(e, \mathbf{x}, y) holds for some y (coding a computation sequence) and a primitive recursive function U(y) = z extracts the output.^[38] Here, T functions as an indicator for whether \phi_e(\mathbf{x}) is defined, with the existential quantifier \exists y \, T(e, \mathbf{x}, y) marking the domain of \phi_e; if no such y exists, the function diverges (undefined).^[38] Introduced in Kleene's 1938 work, this representation unifies partial recursiveness and underscores undecidability, as the halting predicate derived from T mirrors the non-recursive nature of \chi_K.^[38]

In Fuzzy Set Theory

In fuzzy set theory, the classical indicator function of a crisp set, which assigns binary values of 0 or 1 to elements based on membership, serves as a special case of the more general membership function \mu_A: X \to [0,1], where \mu_A(x) = 1_A(x) for all x \in X. This extension allows for degrees of membership that reflect partial belonging, accommodating vagueness and uncertainty inherent in natural language and real-world phenomena.^[42] Lotfi A. Zadeh introduced fuzzy sets in 1965 as a generalization of classical set theory, replacing the binary characteristic (indicator) function with a continuous membership function that maps elements to the unit interval [0,1], thereby enabling the representation of imprecise concepts. This framework unifies and extends traditional indicators by treating crisp sets as fuzzy sets where membership is either fully 0 or 1, while allowing intermediate values for fuzzy sets to model gradations of belonging.^[42] Fuzzy set operations build on these membership functions, adapting classical set theory to handle partial memberships. The intersection of two fuzzy sets A and B is commonly defined using the minimum: \mu_{A \cap B}(x) = \min(\mu_A(x), \mu_B(x)), though the algebraic product \mu_A(x) \cdot \mu_B(x) is also used as a t-norm alternative for probabilistic interpretations. The union is defined via the maximum: \mu_{A \cup B}(x) = \max(\mu_A(x), \mu_B(x)), or the probabilistic sum \mu_A(x) + \mu_B(x) - \mu_A(x) \cdot \mu_B(x) to avoid overcounting overlap. The complement of a fuzzy set A is given by \mu_{\overline{A}}(x) = 1 - \mu_A(x), preserving the duality with crisp complements while allowing graded negation. These operations, rooted in Zadeh's min-max definitions, form the basis for fuzzy logic and have been extended through t-norm and t-conorm families for broader applicability.^[42] Early applications of fuzzy sets, leveraging these generalized indicators, emerged in pattern recognition for classifying ambiguous data patterns and in control theory for managing nonlinear systems with imprecise inputs, such as fuzzy controllers in industrial processes. Zadeh's work highlighted potential in pattern discrimination and information processing, paving the way for subsequent developments in decision systems.^[42]^[43]

Extensions and Approximations

Smooth Approximations

The indicator function $1_A of a set A \subseteq \mathbb{R}^n is discontinuous along the boundary \partial A, rendering it non-differentiable and unsuitable for settings requiring smooth functions, such as the classical theory of partial differential equations (PDEs) where coefficients or initial data must possess sufficient regularity for existence and uniqueness results via methods like Galerkin approximations or maximum principles. Similarly, in optimization problems, the non-smoothness of indicators complicates the computation of gradients or subgradients, hindering the application of efficient algorithms like gradient descent or interior-point methods. Smooth approximations address these issues by replacing $1_A with C^\infty functions that converge to it pointwise or in appropriate norms as a smoothness parameter tends to zero, enabling analytical tractability and numerical stability while preserving essential properties like the integral value \int 1_A = |A|. A primary method for obtaining smooth approximations involves mollification, where the indicator $1_A is convolved with a smooth mollifier kernel \phi_\epsilon(x) = \epsilon^{-n} \phi(x/\epsilon). Here, \phi \in C_c^\infty(\mathbb{R}^n) is a standard mollifier satisfying \int_{\mathbb{R}^n} \phi(x) \, dx = 1, \phi(x) \geq 0, and \operatorname{supp}(\phi) \subseteq B(0,1), with \epsilon > 0 controlling the scale of smoothing. The resulting approximation is u_\epsilon(x) = (1_A * \phi_\epsilon)(x) = \int_{\mathbb{R}^n} 1_A(y) \phi_\epsilon(x - y) \, dy = \int_{A \cap B(x,\epsilon)} \phi_\epsilon(x - y) \, dy, which is C^\infty and satisfies $0 \leq u_\epsilon(x) \leq 1. As \epsilon \to 0^+, u_\epsilon \to 1_A pointwise almost everywhere, and in the L^1(\mathbb{R}^n) norm for Lebesgue measurable A with finite measure, by the density of continuous compactly supported functions in L^1 and properties of approximate identities. This convergence extends to L^p norms for $1 \leq p < \infty when |A| < \infty, with explicit error estimates like \|u_\epsilon - 1_A\|_{L^p} \leq C \epsilon^{1/p} |\partial A| near the boundary, where C depends on \phi and p. Explicit closed-form constructions often employ sigmoid or hyperbolic tangent functions to approximate the Heaviside step function H(t) = 1_{(0,\infty)}(t), which generates indicators for half-spaces; indicators for general sets can then be built via combinations. A common sigmoid approximation is S_k(t) = \frac{1}{1 + e^{-k t}} for large k > 0, satisfying S_k(t) \to H(t) pointwise as k \to \infty, with S_k'(t) = k S_k(t) (1 - S_k(t)) providing a smooth transition zone of width O(1/k). For the indicator of an interval [a, b] in one dimension, a product form $1_{[a,b]}^k(x) \approx S_k(x - a) (1 - S_k(x - b)) yields a C^\infty approximation converging uniformly on compact sets away from \{a, b\}. Alternatively, using the hyperbolic tangent, \frac{1 + \tanh(k (x - c))}{2} approximates H(x - c) with similar properties, as \tanh(z) = 2 S_{2k}(z) - 1 relates directly to the logistic sigmoid; convergence follows from the monotone convergence theorem, with uniform rates on compacts given by |S_k(t) - H(t)| \leq e^{-k |t| + O(\log k)} for |t| \geq \delta > 0. Convergence theorems for these approximations are well-established in functional analysis. For mollifiers, if A has a smooth boundary, u_\epsilon converges to $1_A uniformly on compact subsets of \mathbb{R}^n \setminus \partial A, and in Sobolev spaces W^{s,p}(\mathbb{R}^n) for s < 1 - 1/p (with p \geq 1), the embedding ensures higher regularity of u_\epsilon while controlling the approximation error by the Minkowski content of \partial A. Sigmoid-based approximations exhibit analogous behavior: for the Heaviside, S_k \to H in the sense of distributions, and for bounded variation functions involving indicators, the total variation \|S_k\|_{BV} \to \|H\|_{BV} = 1 as k \to \infty, facilitating applications in calculus of variations. These results underpin the use of smooth indicators in numerical schemes for PDEs, such as level-set methods, where the approximation parameter is adaptively chosen to balance accuracy and computational cost.

Generalizations to Measures and Beyond

In measure theory, simple functions are defined as finite linear combinations of indicator functions over measurable sets, typically expressed as \phi = \sum_{i=1}^n a_i \mathbf{1}_{A_i}, where each A_i is a measurable set with finite measure and a_i \in \mathbb{R}.^[11] These functions form a dense subspace in the L^p(\mu) spaces for $1 \leq p < \infty on a \sigma-finite measure space (\Omega, \mathcal{F}, \mu), meaning any f \in L^p(\mu) can be approximated arbitrarily closely in the L^p norm by simple functions.^[44] This density property is crucial for extending integrals and establishing completeness of L^p spaces.^[45] The Lebesgue integral is initially defined for nonnegative simple functions as \int \phi \, d\mu = \sum_{i=1}^n a_i \mu(A_i), providing a foundational step for integrating more general measurable functions.^[46] For a nonnegative measurable function f, the integral is then constructed as the supremum of integrals over simple functions \phi \leq f, ensuring the Lebesgue integral coincides with the Riemann integral where the latter exists and extends naturally to abstract measure spaces.^[47] This approach underpins the theory of integration with respect to arbitrary measures, enabling the handling of functions that are not Riemann-integrable, such as the Dirichlet function. Beyond measure theory, indicator functions generalize to abstract categorical settings, particularly in topos theory, where the subobject classifier \Omega serves an analogous role to the power object of sets. In an elementary topos, for any object X and subobject S \hookrightarrow X, there exists a unique characteristic morphism \chi_S: X \to \Omega that "indicates" membership in S, generalizing the classical indicator function and enabling internal logic within the category.^[48] In topology, indicator functions of clopen sets—subsets that are both open and closed—are precisely the continuous functions valued in \{0,1\}, facilitating the study of disconnected spaces and measures on compact Hausdorff spaces.^[49] In modern applications, indicator functions appear in optimal transport theory, where they define constraints for transport plans in the Kantorovich formulation of the Wasserstein distance, measuring discrepancies between probability measures via minimal cost couplings.^[50] Similarly, in machine learning, the 0-1 loss function, defined as L(y, t) = \mathbf{1}_{\{y \neq t\}} for predicted label y and true label t, quantifies classification errors but is often surrogated by convex losses due to its non-differentiability.^[51]

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An indicator random variable is a random variable that maps every outcome to either 0 or 1. Indicator random variables are also called Bernoulli variables.
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[PDF] 07: Variance, Bernoulli, Binomial
Jan 22, 2020 · def A Bernoulli random variable XX maps “success” to 1 and “failure” to 0. Other names: indicator random variable, boolean random variable.
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[PDF] Chapter 8: The Poisson Paradigm - Sites@Duke Express
Nov 18, 2022 · Poisson Paradigm. When X is the sum of many rare indicator “mostly independent” random variables, and µ = E[X], we would like to say that X ...
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[PDF] Poisson approximations
The Bin(n, p) can be thought of as the distribution of a sum of independent indicator random variables X1 + ... + Xn, with {Xi = 1} denoting a head on the ...
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Poisson Approximation Using the Stein-Chen Method and Coupling
The bound is derived as a special case of a result obtained by the Stein-Chen method for sums of dependent Bernoulli random variables. This general result ...
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[PDF] Chapter 4 - RENEWAL PROCESSES - MIT OpenCourseWare
Recall that a renewal process is an arrival process in which the interarrival intervals are positive,1 independent and identically distributed (IID) random ...
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[PDF] 1 Introduction to Renewal Theory - Columbia University
But for a general renewal process, the distribution of A(t) is complicated and depends on the time t. A(s)ds = E(X2) 2E(X) w.p.1.
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Hilbert's Program - Stanford Encyclopedia of Philosophy
Jul 31, 2003 · Gödel's incompleteness theorems showed that Hilbert's optimism was undue. In September 1930, Kurt Gödel announced his first incompleteness ...
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Recursive Functions - Stanford Encyclopedia of Philosophy
Apr 23, 2020 · A primary problem in the theory of recursively enumerable sets is the problem of determining the degrees of unsolvability of the unsolvable ...
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Gödel Numbering - Stanford Encyclopedia of Philosophy
A key method in the usual proofs of the first incompleteness theorem is the arithmetization of the formal language, or Gödel numbering.
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https://plato.stanford.edu/entries/goedel-incompleteness/sup1.html
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Fuzzy sets - ScienceDirect.com
A fuzzy set is a class of objects with a continuum of membership grades, characterized by a membership function ranging from zero to one.
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[PDF] Zadeh L A. Fuzzy sets. Inform. Contr. 8:338-53, 1965.
Nov 24, 1980 · Viewed as such, the theory of fuzzy sets is finding many applications in a wide variety of fields ranging from industrial process control to ...
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[PDF] Lp spaces - UC Davis Math
Density theorems enable us to prove properties of Lp functions by proving them for functions in a dense subspace and then extending the result by continuity.
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[PDF] lebesgue measure and l2 space. - UChicago Math
This theorem shows that any measurable function can be approximated by simple functions, and therefore also by linear combinations of simple functions. We will.
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245A, Notes 2: The Lebesgue integral | What's new - Terry Tao
Sep 19, 2010 · -measurable function is as the pointwise limit of simple functions (finite linear combinations of indicator functions of measurable sets).
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[PDF] 18.102 S2021 Lecture 10. Simple Functions - MIT OpenCourseWare
Mar 23, 2021 · Our first step is to start with Lebesgue integrals of nonnegative functions (to avoid things like ∞−∞, and because as we've just seen, knowing ...
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Is defining measures as functionals ever insufficiently general in ...
Jan 28, 2024 · Using the linear functionals on C(X) approach, one can only take the measure of a set S whose indicator function χS is continuous, i.e. a clopen ...Why is a topology made up of 'open' sets? - MathOverflowCharacterization of state spaces of Boolean algebras - MathOverflowMore results from mathoverflow.net
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[PDF] Optimal transport, old and new - Ceremade
Sep 27, 2006 · The first part is devoted to a qualitative description of optimal transport; the second part discusses the use of optimal transport in ...
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[PDF] Pattern Recognition and Machine Learning - Microsoft
A common choice of loss function in regression problems is the squared loss given by L(t, y(x)) = {y(x) − t}2. In this case, the expected loss can be written.