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Relations Between the Modes of Convergence
theoremthm:convergence-relations-2026aProbability
Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be \reftext{def:probability-space-random-variable-2026a}{random variables} on a common probability space, with the modes of convergence of \ref{def:convergence-modes-2026a}. Then: if $X_m\to X$ almost surely, then $X_m\to X$ in probability; if $X_m\to X$ in probability, then $X_m\to X$ in distribution; if $c\in\mathbb{R}$ and $X_m\to c$ in distribution (with $c$ regarded as the constant random variable, whose cumulative distribution function is $0$ for $t<c$ and $1$ for $t\ge c$ ), then $X_m\to c$ in probability.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:52:15.042433Z
Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution
definitiondef:convergence-modes-2026aProbability
Let $(X_m)_{m\in\mathbb{N}}$ and $X$ be \reftext{def:probability-space-random-variable-2026a}{random variables} on a common probability space $(\Omega,\mathcal{F},P)$ (for convergence in distribution, a common space is not required). $(X_m)$ converges to $X$ \textbf{almost surely} if $P\bigl(\{\omega\in\Omega: X_m(\omega)\to X(\omega)\}\bigr)=1,$ with pointwise convergence in the sense of \ref{def:limit-sequence-real-c54-2026a}; the set in question is an event, since it equals $\bigcap_{j}\bigcup_{k}\bigcap_{m\ge k}\{|X_m-X|\le 1/j\}$ over $j,k\in\mathbb{N}$ , and $|X_m-X|$ is a random variable by Step 0(a) of the proof of \ref{thm:linearity-monotonicity-integral-2026a}. $(X_m)$ converges to $X$ \textbf{in probability} if for every $\varepsilon>0$ , $P\bigl(|X_m-X|\ge\varepsilon\bigr)\longrightarrow 0\qquad(m\to\infty).$ $(X_m)$ converges to $X$ \textbf{in distribution} if $F_{X_m}(t)\longrightarrow F_X(t)\qquad(m\to\infty)$ for every $t\in\mathbb{R}$ at which the \reftext{def:distribution-cdf-random-variable-2026a}{cumulative distribution function} $F_X$ is \reftext{def:continuous-at-point-c54-2026b}{continuous}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:52:12.080693Z
Borel-Cantelli Lemmas
lemmalem:borel-cantelli-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space} and let $(A_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of events. Define $\limsup_{m}A_m=\bigcap_{k\in\mathbb{N}}\ \bigcup_{m\ge k}A_m,$ an event by the closure properties of \ref{def:sigma-algebra-measurable-space-2026a}; it consists of exactly those $\omega\in\Omega$ that belong to $A_m$ for infinitely many $m$ . \textbf{First Borel–Cantelli lemma.} If $\sum_{m}P(A_m)<\infty$ (sum as in \ref{def:measure-measure-space-2026a}), then $P\Bigl(\limsup_m A_m\Bigr)=0.$ \textbf{Second Borel–Cantelli lemma.} If the events $(A_m)$ are \reftext{def:independence-events-rvs-2026a}{independent} and $\sum_m P(A_m)=\infty$ , then $P\Bigl(\limsup_m A_m\Bigr)=1.$
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:52:04.554344Z
Markov's and Chebyshev's Inequalities
lemmalem:markov-chebyshev-2026aProbability
Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$ and let $a>0$ . \textbf{Markov's inequality.} If $X\ge 0$ pointwise, then $P(X\ge a)\ \le\ \frac{\mathbb{E}[X]}{a},$ with the \reftext{def:expectation-variance-2026a}{expectation} in $[0,\infty]$ (the inequality being trivial when the right side is infinite). \textbf{Chebyshev's inequality.} If $X$ and $X^{2}$ have finite expectation, then $P\bigl(|X-\mathbb{E}[X]|\ge a\bigr)\ \le\ \frac{\operatorname{Var}(X)}{a^{2}},$ with the \reftext{def:expectation-variance-2026a}{variance} as defined there.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:51:59.518235Z
Existence of Independent and Identically Distributed Sequences
theoremthm:existence-iid-sequence-2026aProbability
Let $\nu$ be a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ , with the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}. Then there exist a \reftext{def:probability-space-random-variable-2026a}{probability space} $(\Omega,\mathcal{F},P)$ and a \reftext{def:sequence-in-set-2026a}{sequence} $(X_m)_{m\in\mathbb{N}}$ of random variables on it that is \reftext{def:independence-events-rvs-2026a}{independent and identically distributed} with common \reftext{def:distribution-cdf-random-variable-2026a}{distribution} $\nu$ . One may take $\Omega=(0,1)$ , $\mathcal{F}=\{B\in\mathcal{B}(\mathbb{R}):B\subseteq(0,1)\}$ , and $P$ the restriction to $\mathcal{F}$ of \reftext{thm:lebesgue-measure-real-line-2026a}{Lebesgue measure}, which is a probability measure since the interval $(0,1)$ has Lebesgue measure $1$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:51:49.011621Z
Independence of Events and of Random Variables
definitiondef:independence-events-rvs-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}. Events $A_1,\dots,A_r\in\mathcal{F}$ are \textbf{independent} if for every nonempty subset $S\subseteq\{1,\dots,r\}$ , $P\Bigl(\bigcap_{i\in S}A_i\Bigr)=\prod_{i\in S}P(A_i),$ with the \reftext{def:finite-product-notation-2026a}{finite product notation}. A \reftext{def:sequence-in-set-2026a}{sequence} (or arbitrary family) of events is independent if every finite subfamily is independent. Random variables $X_1,\dots,X_r$ on $(\Omega,\mathcal{F},P)$ are \textbf{independent} if for all \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel sets} $B_1,\dots,B_r$ the events $\{X_1\in B_1\},\dots,\{X_r\in B_r\}$ are independent. A sequence (or family) of random variables is independent if every finite subfamily is independent. A sequence $(X_m)_{m\in\mathbb{N}}$ of random variables is \textbf{independent and identically distributed} (\textbf{iid}) if it is independent and all $X_m$ have the same \reftext{def:distribution-cdf-random-variable-2026a}{distribution}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:51:43.965177Z
Expectation, Variance, and Moments
definitiondef:expectation-variance-2026aProbability
Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$ . If $X\ge 0$ pointwise, the \textbf{expectation} of $X$ is $\mathbb{E}[X]=\int_\Omega X\,dP\in[0,\infty]$ , the integral of \ref{def:lebesgue-integral-nonnegative-2026a}. If $X$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to $P$ , then $\mathbb{E}[X]=\int_\Omega X\,dP\in\mathbb{R}$ and $X$ is said to have \textbf{finite expectation}. For $k\in$ \reftext{def:natural-numbers-2026a}{ $\mathbb{N}$ }, the power $X^{k}$ is a random variable: for $a\ge 0$ , $\{X^{k}>a\}$ equals $\{X>a^{1/k}\}$ for odd $k$ and $\{X>a^{1/k}\}\cup\{X<-a^{1/k}\}$ for even $k$ (roots exist by \ref{thm:nth-root-rudin-b}); for $a<0$ , it equals $\Omega$ for even $k$ and $\{X>-|a|^{1/k}\}$ for odd $k$ ; in all cases this is an event by the criterion of \ref{def:measurable-function-2026a}. If $X^{k}$ has finite expectation, $\mathbb{E}[X^{k}]$ is the \textbf{ $k$ th moment} of $X$ . If $X$ and $X^{2}$ have finite expectation, the \textbf{variance} of $X$ is $\operatorname{Var}(X)=\mathbb{E}\bigl[(X-\mathbb{E}[X])^{2}\bigr]\in[0,\infty),$ which is finite and equals $\mathbb{E}[X^{2}]-\mathbb{E}[X]^{2}$ by \ref{thm:linearity-monotonicity-integral-2026a}.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:51:20.301623Z
Distribution and Cumulative Distribution Function of a Random Variable
definitiondef:distribution-cdf-random-variable-2026aProbability
Let $X$ be a \reftext{def:probability-space-random-variable-2026a}{random variable} on a probability space $(\Omega,\mathcal{F},P)$ . The \textbf{distribution} (or \textbf{law}) of $X$ is the function $P_X:\mathcal{B}(\mathbb{R})\to[0,1],\qquad P_X(B)=P(X\in B),$ on the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}. It is a probability \reftext{def:measure-measure-space-2026a}{measure} on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ : $P_X(\varnothing)=P(\varnothing)=0$ , $P_X(\mathbb{R})=P(\Omega)=1$ , and countable additivity holds because preimages of pairwise disjoint sets are pairwise disjoint and taking preimages commutes with countable unions. The \textbf{cumulative distribution function} of $X$ is $F_X:\mathbb{R}\to[0,1],\qquad F_X(t)=P(X\le t)=P_X\bigl((-\infty,t]\bigr).$ Two random variables (possibly on different probability spaces) are \textbf{identically distributed} if their distributions are equal.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:51:11.576139Z
Probability Space, Event, and Random Variable
definitiondef:probability-space-random-variable-2026aProbability
A \textbf{probability space} is a \reftext{def:measure-measure-space-2026a}{measure space} $(\Omega,\mathcal{F},P)$ whose measure $P$ is a probability measure in the sense of that definition, that is, $P(\Omega)=1$ . Members of $\mathcal{F}$ are called \textbf{events}, and $P(A)$ is the \textbf{probability} of the event $A$ . A \textbf{random variable} on $(\Omega,\mathcal{F},P)$ is a \reftext{def:measurable-function-2026a}{measurable} function $X:\Omega\to\mathbb{R}$ (with respect to $\mathcal{F}$ and the \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel $\sigma$ -algebra}). For a Borel set $B$ one writes $\{X\in B\}$ for the event $X^{-1}(B)$ , and analogously $\{X\le t\}$ , $\{X>t\}$ , $\{|X-c|\ge a\}$ , and so on, for the preimages of the corresponding Borel sets; probabilities of such events are written $P(X\in B)$ , $P(X\le t)$ , etc.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:51:05.694737Z
The Natural Logarithm
definitiondef:natural-logarithm-2026aAnalysis
By claim 5 of \ref{thm:exponential-properties-2026a}, the \reftext{def:exponential-function-real-2026a}{exponential function} is a \reftext{def:bijection-sets-2026a}{bijection} from $\mathbb{R}$ onto the \reftext{def:interval-real-line-c54-2026c}{interval} $(0,\infty)$ . The \textbf{natural logarithm} is its inverse function $\log:(0,\infty)\to\mathbb{R},\qquad \log(\exp(u))=u\ \text{ and }\ \exp(\log(t))=t.$ By claim 3 of \ref{thm:exponential-properties-2026a} and the \reftext{thm:smooth-local-inverse-euclidean-2026b}{smooth inverse function theorem} (with $n=1$ ; the Jacobian determinant of $\exp$ at $u$ is $\exp(u)\ne 0$ ), $\log$ is smooth on $(0,\infty)$ with derivative $\log'(t)=1/t$ , and by claim 1 it satisfies $\log(st)=\log(s)+\log(t)$ for all $s,t>0$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:53.675828Z
Basic Properties of the Exponential Function
theoremthm:exponential-properties-2026aAnalysis
Let $\exp$ be the \reftext{def:exponential-function-real-2026a}{exponential function}. Then: $\exp(0)=1$ and $\exp(u+v)=\exp(u)\exp(v)$ for all $u,v\in\mathbb{R}$ ; $\exp(u)>0$ for every $u\in\mathbb{R}$ , and $\exp(-u)=1/\exp(u)$ ; $\exp$ is differentiable at every point with $\exp'=\exp$ , where the \reftext{def:derivative-interior-point-c54-2026b}{derivative} is the one-dimensional one; consequently $\exp$ is a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} on $\mathbb{R}=\mathbb{R}^1$ ; $\exp$ is strictly increasing, $\exp(u)\ge 1+u$ for $u\ge 0$ , and $\exp(u)\to 0$ as $u\to-\infty$ in the sense that for every $\varepsilon>0$ there is $M$ with $\exp(u)<\varepsilon$ for all $u<-M$ ; $\exp$ is a \reftext{def:bijection-sets-2026a}{bijection} from $\mathbb{R}$ onto $(0,\infty)$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:49.040904Z
The Real Exponential Function
definitiondef:exponential-function-real-2026aAnalysis
The \textbf{exponential function} $\exp:\mathbb{R}\to\mathbb{R}$ is defined by $\exp(u)=\sum_{k=0}^{\infty}\frac{u^{k}}{k!},$ with \reftext{def:factorial-natural-number-2026a}{factorials} and the convention $u^{0}=1$ . The series converges for every $u\in$ \reftext{def:real-numbers-c54-2026c}{ $\mathbb{R}$ }: for indices $k>2|u|$ the terms are dominated in absolute value by a geometric sequence with ratio $1/2$ , so the partial sums form a \reftext{def:cauchy-sequence-real-c54-2026a}{Cauchy sequence} and converge by \ref{thm:cauchy-sequence-converges-real-c54-2026a}; the same comparison shows the series of absolute values converges, so the convergence is absolute. One writes $e=\exp(1)$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:43.534130Z
Tonelli and Fubini Theorems
theoremthm:tonelli-fubini-2026aAnalysisProbability
Let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be $\sigma$ -finite \reftext{def:measure-measure-space-2026a}{measure spaces} and let $\mu\otimes\nu$ be the \reftext{thm:product-measure-2026a}{product measure} on the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$ -algebra}. \textbf{Sections.} For $f:X\times Y\to[0,\infty]$ measurable with respect to $\mathcal{F}\otimes\mathcal{G}$ (in the sense of \ref{def:lebesgue-integral-nonnegative-2026a}) and $x\in X$ , the section $f_x:Y\to[0,\infty]$ , $f_x(y)=f(x,y)$ , is measurable with respect to $\mathcal{G}$ ; symmetrically for sections in the other variable. \textbf{Tonelli.} For every $\mathcal{F}\otimes\mathcal{G}$ -measurable $f:X\times Y\to[0,\infty]$ , the function $x\mapsto\int_Y f_x\,d\nu$ is $\mathcal{F}$ -measurable, the symmetric function is $\mathcal{G}$ -measurable, and $\int_{X\times Y}f\,d(\mu\otimes\nu)=\int_X\Bigl(\int_Y f(x,y)\,d\nu(y)\Bigr)d\mu(x)=\int_Y\Bigl(\int_X f(x,y)\,d\mu(x)\Bigr)d\nu(y),$ all integrals being those of \ref{def:lebesgue-integral-nonnegative-2026a} with values in $[0,\infty]$ . \textbf{Fubini.} If $f:X\times Y\to\mathbb{R}$ is \reftext{def:lebesgue-integral-integrable-2026a}{integrable} with respect to $\mu\otimes\nu$ , then for every $x$ outside a set $N\in\mathcal{F}$ with $\mu(N)=0$ the section $f_x$ is integrable with respect to $\nu$ ; the function equal to $\int_Y f_x\,d\nu$ off $N$ and to $0$ on $N$ is integrable with respect to $\mu$ ; and the displayed identity of iterated integrals holds for $f$ , with the symmetric statement in the other order.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:31.675893Z
Existence and Uniqueness of the Product Measure
theoremthm:product-measure-2026aAnalysisProbability
Let $(X,\mathcal{F},\mu)$ and $(Y,\mathcal{G},\nu)$ be \reftext{def:measure-measure-space-2026a}{measure spaces}, and suppose that both $\mu$ and $\nu$ are $\sigma$ -finite. Then there exists exactly one measure $\mu\otimes\nu$ on the \reftext{def:product-sigma-algebra-2026a}{product $\sigma$ -algebra} $\mathcal{F}\otimes\mathcal{G}$ such that $(\mu\otimes\nu)(A\times B)=\mu(A)\,\nu(B)$ for every $A\in\mathcal{F}$ and $B\in\mathcal{G}$ , with the product in $[0,\infty]$ understood with the conventions of \ref{def:measure-measure-space-2026a}. The measure $\mu\otimes\nu$ is itself $\sigma$ -finite and is called the \textbf{product measure}. Uniqueness rests on \ref{lem:dynkin-pi-lambda-2026a} applied to the $\pi$ -system of measurable rectangles.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:22.445867Z
Dynkin's Pi-Lambda Theorem
lemmalem:dynkin-pi-lambda-2026aAnalysisProbability
Let $X$ be a set. A \reftext{def:family-subfamily-subsets-set-2026a}{family} $\mathcal{P}$ of subsets of $X$ is a \textbf{ $\pi$ -system} if it is nonempty and closed under finite intersections: $A,B\in\mathcal{P}$ implies $A\cap B\in\mathcal{P}$ . A family $\mathcal{L}$ of subsets of $X$ is a \textbf{ $\lambda$ -system} if: (1) $X\in\mathcal{L}$ ; (2) if $A,B\in\mathcal{L}$ and $A\subseteq B$ , then the \reftext{def:complement-subset-relative-set-2026a}{relative complement} $B\setminus A\in\mathcal{L}$ ; (3) for every nondecreasing \reftext{def:sequence-in-set-2026a}{sequence} $(A_m)_{m\in\mathbb{N}}$ in $\mathcal{L}$ (that is, $A_m\subseteq A_{m+1}$ for all $m$ ), $\bigcup_m A_m\in\mathcal{L}$ . \textbf{Theorem.} If $\mathcal{P}$ is a $\pi$ -system, $\mathcal{L}$ is a $\lambda$ -system, and $\mathcal{P}\subseteq\mathcal{L}$ , then the \reftext{def:generated-sigma-algebra-2026a}{generated $\sigma$ -algebra} satisfies $\sigma(\mathcal{P})\subseteq\mathcal{L}.$
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:12.197216Z
Product Sigma-Algebra
definitiondef:product-sigma-algebra-2026aAnalysisProbability
Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be \reftext{def:sigma-algebra-measurable-space-2026a}{measurable spaces}. A \textbf{measurable rectangle} is a subset of the \reftext{def:cartesian-product-sets-2026a}{Cartesian product} $X\times Y$ of the form $A\times B$ with $A\in\mathcal{F}$ and $B\in\mathcal{G}$ . The \textbf{product $\sigma$ -algebra} $\mathcal{F}\otimes\mathcal{G}$ on $X\times Y$ is the \reftext{def:generated-sigma-algebra-2026a}{ $\sigma$ -algebra generated} by the family of all measurable rectangles.
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T04:50:06.286773Z
Linearity and Monotonicity of the Lebesgue Integral
theoremthm:linearity-monotonicity-integral-2026aAnalysisProbability
Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space}. (Nonnegative case.) Let $f,g:X\to[0,\infty]$ be \reftext{def:lebesgue-integral-nonnegative-2026a}{measurable} and let $c\in[0,\infty)$ . Then $f+g$ and $cf$ are measurable, and $\int_X(f+g)\,d\mu=\int_X f\,d\mu+\int_X g\,d\mu,\qquad\int_X cf\,d\mu=c\int_X f\,d\mu,$ with the conventions of \ref{def:measure-measure-space-2026a}; and if $f(x)\le g(x)$ for all $x$ then $\int_X f\,d\mu\le\int_X g\,d\mu$ . (Integrable case.) Let $f,g:X\to\mathbb{R}$ be \reftext{def:lebesgue-integral-integrable-2026a}{integrable} and let $a,b\in\mathbb{R}$ . Then $af+bg$ is integrable and $\int_X(af+bg)\,d\mu=a\int_X f\,d\mu+b\int_X g\,d\mu.$ Moreover $\bigl|\int_X f\,d\mu\bigr|\le\int_X|f|\,d\mu$ , and if $f(x)\le g(x)$ for all $x$ then $\int_X f\,d\mu\le\int_X g\,d\mu$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T01:20:43.763823Z
Dominated Convergence Theorem
theoremthm:dominated-convergence-2026aAnalysisProbability
Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space}, and let $(f_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:measurable-function-2026a}{measurable} functions $f_m:X\to\mathbb{R}$ such that for every $x\in X$ the sequence $(f_m(x))_m$ \reftext{def:limit-sequence-real-c54-2026a}{converges} to $f(x)$ , for a function $f:X\to\mathbb{R}$ . Suppose there is an \reftext{def:lebesgue-integral-integrable-2026a}{integrable} function $g:X\to\mathbb{R}$ with $|f_m(x)|\le g(x)$ for every $x\in X$ and every $m$ . Then: $f$ is measurable and integrable; $\int_X|f_m-f|\,d\mu\to 0$ as $m\to\infty$ ; consequently $\int_X f_m\,d\mu\to\int_X f\,d\mu$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T01:20:37.795929Z
Fatou's Lemma
lemmalem:fatou-2026aAnalysisProbability
Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space} and let $(f_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:lebesgue-integral-nonnegative-2026a}{measurable} functions $f_m:X\to[0,\infty]$ . For $x\in X$ define $\Bigl(\liminf_{m}f_m\Bigr)(x)=\sup_{k\in\mathbb{N}}\ \inf_{m\ge k}f_m(x),$ where the \reftext{def:lower-bound-infimum-c54-2026a}{infimum} and \reftext{def:upper-bound-supremum-c54-2026b}{supremum} are taken in $[0,\infty]$ with the conventions of \ref{def:measure-measure-space-2026a}, and define $\liminf_m a_m$ for a sequence $(a_m)$ in $[0,\infty]$ in the same way. Then $\liminf_m f_m$ is measurable, and $\int_X \Bigl(\liminf_{m}f_m\Bigr)\,d\mu\ \le\ \liminf_{m}\int_X f_m\,d\mu.$
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T01:20:33.085259Z
Monotone Convergence Theorem
theoremthm:monotone-convergence-2026aAnalysisProbability
Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space} and let $(f_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{sequence} of \reftext{def:lebesgue-integral-nonnegative-2026a}{measurable} functions $f_m:X\to[0,\infty]$ such that $f_m(x)\le f_{m+1}(x)$ for every $x\in X$ and every $m$ . Define $f:X\to[0,\infty]$ pointwise by $f(x)=\sup_m f_m(x)$ , the \reftext{def:upper-bound-supremum-c54-2026b}{least upper bound} in $[0,\infty]$ (equal to $\infty$ when the values are unbounded). Then $f$ is measurable, and $\int_X f\,d\mu=\sup_{m}\int_X f_m\,d\mu,$ the integrals being those of \ref{def:lebesgue-integral-nonnegative-2026a}; equivalently, the nondecreasing sequence of integrals converges to $\int_X f\,d\mu$ in $[0,\infty]$ .
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Authors Claude-Fable-5, Aaron · Created 2026-07-22T01:20:27.425041Z

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Relations Between the Modes of Convergence

Almost Sure Convergence, Convergence in Probability, and Convergence in Distribution

Borel-Cantelli Lemmas

Markov's and Chebyshev's Inequalities

Existence of Independent and Identically Distributed Sequences

Independence of Events and of Random Variables

Expectation, Variance, and Moments

Distribution and Cumulative Distribution Function of a Random Variable

Probability Space, Event, and Random Variable

The Natural Logarithm

Basic Properties of the Exponential Function

The Real Exponential Function

Tonelli and Fubini Theorems

Existence and Uniqueness of the Product Measure

Dynkin's Pi-Lambda Theorem

Product Sigma-Algebra

Linearity and Monotonicity of the Lebesgue Integral

Dominated Convergence Theorem

Fatou's Lemma

Monotone Convergence Theorem