The Integral of Nonnegative Measurable Functions

# The Integral of Nonnegative Measurable Functions

Recall from The Integral of Nonnegative Simple Functions page that if $(X, \mathcal A, \mu)$ is a complete measure space then the integral of a nonnegative simple function $\varphi$ defined on a measurable set $E$ is as follows. For $\psi(x) = 0$ we define $\displaystyle{\int_E \psi(x) \: d \mu = 0}$. And if $a_1, a_2, ..., a_n > 0$ are the positive values in the range of $\varphi$ with $\varphi(x) = a_k$ if and only if $x \in E_k \subseteq E$ then:

(1)
\begin{align} \quad \int_E \varphi(x) \: d \mu = \sum_{k=1}^{n} a_k \mu (E_k) \end{align}

We now define the integral of nonnegative measurable functions for a general complete measure space.

 Definition: Let $(X, \mathcal A, \mu)$ be a complete measure space. The Integral of a Nonnegative Measurable Function $f$ defined on a measurable set $E$ is $\displaystyle{\int_E f(x) \: d \mu = \sup \left \{ \int_E \varphi(x) \: d \mu : \varphi \: \mathrm{is \: a \: simple \: function}, \: 0 \leq \varphi(x) \leq f(x) \right \}}$.

Once again, many of the basic theorems for general integrals hold.

 Theorem 1 (Linearity of the Integral of Nonnegative Simple Functions): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ and $g$ be nonnegative measurable functions defined on a measurable set $E$. Then for all $\alpha, beta \in \mathbb{R}$ with $\alpha, \beta \geq 0$ we have that $\displaystyle{\int_E (\alpha f(x) + \beta g(x)) \: d \mu = \alpha \int_E f(x) \: d \mu + \beta \int_E g(x) \: d \mu}$.

We will prove Theorem 1 later on The Linearity Property of the Integral of Nonnegative Measurable Functions page since we will first need to prove a Monotone converge theorem for general measure spaces.

 Theorem 2 (Monotonicity of the Integral of Nonnegative Simple Functions): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ and $g$ be nonnegative measurable functions defined on a measurable set $E$. If $f(x) \leq g(x)$ on $E$ then $\displaystyle{\int_E f(x) \: d \mu \leq \int_E g(x) \: d \mu}$.
 Theorem 3 (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 defined on the measurable sets $A$ and $B$. If $A$ and $B$ are mutually disjoint then $\displaystyle{\int_{A \cup B} f(x) \: d \mu = \int_A f(x) \: d \mu + \int_B f(x) \: d \mu}$.

We will prove Theorem 3 later on The Additivity Over Domains of Integration Property of Nonnegative Measurable Functions page.

 Theorem 4 (The Countable Additivity Over Domains of Integration): Let $(X, \mathcal A, \mu)$ be a complete measure space and let $f$ be a nonnegative measurable function defined on the measurable sets $(E_n)_{n=1}^{\infty}$ where $(E_n)_{n=1}^{\infty}$ is a collection of mutually disjoint sets. Then $\displaystyle{\int_{\bigcup_{n=1}^{\infty} E_n} f(x) \: d \mu = \sum_{n=1}^{\infty} \int_{E_n} f(x) \: d \mu}$.