FoldUnfold Table of Contents Measures on Algebras of Sets Measures on Algebras of Sets Definition: Let $X$ be a set and let $\mathcal A$ be an algebra of sets (not necessarily a $\sigma$-algebra) on $X$. A Measure on $\mathcal A$ is a set function $\mu : \mathcal A \to [0, \infty]$ with the following properties: 1) $\mu (\emptyset) = 0$. 2) If $(A_n)_{n=1}^{\infty}$ is … [Read more...]

## Outer Measurable Sets

FoldUnfold Table of Contents Outer Measurable Sets Outer Measurable Sets Recall from the Outer Measures on Measurable Spaces page that if we have the measurable space $(X, \mathcal P(X))$ then an outer measure on this space is a set function $\mu^* : \mathcal P(X) \to [0, \infty]$ with the following properties: 1) $\mu^*(\emptyset) = 0$. 2) If $A$ and $B$ are subsets … [Read more...]

## Outer Measures on Measurable Spaces

FoldUnfold Table of Contents Outer Measures on Measurable Spaces Outer Measures on Measurable Spaces Definition: Let $(X, \mathcal P(X))$ be a measurable space. An Outer Measure on this space is a set function $\mu^* : \mathcal P(X) \to [0, \infty]$ with the following properties: 1) $\mu^*(\emptyset) = 0$. 2) If $A$ and $B$ are subsets of $X$ and $A \subseteq B$ then … [Read more...]

## The Dominated Convergence Theorem for Measurable Functions

FoldUnfold Table of Contents The Dominated Convergence Theorem for Measurable Functions The Dominated Convergence Theorem for Measurable Functions Recall from The Lebesgue Dominated Convergence Theorem that if $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue measurable functions defined on a Lebesgue measurable set $E$ such that: 1) There exists a nonnegative Lebesgue … [Read more...]

## The Comparison Test for Integrability

FoldUnfold Table of Contents The Comparison Test for Integrability The Comparison Test for Integrability Recall from The Comparison Test for Lebesgue Integrability that if $f$ is a Lebesgue measurable function defined on a Lebesgue measurable set $E$ and if there exists a nonnegative Lebesgue measurable function $g$ on $E$ such that: 1) $|f(x)| \leq g(x)$ almost … [Read more...]

## The Integral of Measurable Functions

FoldUnfold Table of Contents The Integral of Measurable Functions The Integral of Measurable Functions Recall that if $f$ is a function and $f^+ = \max \{ f, 0 \}$ and $f^- = \max \{ -f, 0\}$ then: (1) \begin{align} \quad f = f^+ - f^- \end{align} And: (2) \begin{align} \quad |f| = f^+ + f^- \end{align} We have already defined the integral for a nonnegative measurable … [Read more...]

## Integrals of Nonnegative Measurable Functions that Equal Zero

FoldUnfold Table of Contents Integrals of Nonnegative Measurable Functions that Equal Zero Integrals of Nonnegative Measurable Functions that Equal Zero Theorem 1: Let $(X, \mathcal A, \mu)$ be a complete measure space. If $f$ is a nonnegative measurable function defined on a measurable set $E$ and $\displaystyle{\int_E f(x) \: d \mu = 0}$ then $f(x) = 0$ $mu$-almost … [Read more...]

## The Additivity Over Domains of Integration Property of Nonnegative Measurable Functions

FoldUnfold Table of Contents The Additivity Over Domains of Integration Property of Nonnegative Measurable Functions The Additivity Over Domains of Integration Property of Nonnegative Measurable Functions Theorem 1 (The Finite Additivity Over Domains of Integration): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ be a nonnegative measurable function … [Read more...]

## The Linearity Property of the Integral of Nonnegative Measurable Functions

FoldUnfold Table of Contents The Linearity Property of the Integral of Nonnegative Measurable Functions The Linearity Property of the Integral of Nonnegative Measurable Functions Theorem 1 (Linearity of the Integral of Nonnegative Measurable Functions): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ and $g$ be nonnegative measurable functions defined … [Read more...]

## Beppo Levi’s Lemma for Nonnegative Increasing Measurable Functions

FoldUnfold Table of Contents Beppo Levi's Lemma for Nonnegative Increasing Measurable Functions Beppo Levi's Lemma for Nonnegative Increasing Measurable Functions Lemma 1: Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ be an extended nonnegative measurable function defined on a measurable set $E$ such that $\displaystyle{\int_E f(x) \: d \mu . … [Read more...]