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...]