The Lebesgue Integral of Nonnegative Lebesgue Measurable Functions

The Lebesgue Integral of Nonnegative Lebesgue Measurable Functions

We have already defined the Lebesgue integral for simple functions $\varphi$. We have also defined when a bounded function defined on a Lebesgue measurable set $E$ with $m(E) < \infty$ is Lebesgue integrable, and noted that every bounded Lebesgue measurable function defined on a Lebesgue measurable set $E$ with $m(E) < \infty$ is Lebesgue integrable.

We now extend the definition of Lebesgue integrability to nonnegative Lebesgue measurable functions defined on a Lebesgue measurable set $E$. This time we do NOT require the Lebesgue measure of $E$ to be finite.

 Definition: Let $f$ be a nonnegative Lebesgue measurable function defined on a Lebesgue measurable set $E$. Then Lebesgue Integral of $f$ on $E$ is $\displaystyle{\int_E f = \sup \left \{ \int_{E_{\varphi}} \varphi : \varphi \: \mathrm{is \: bounded \: and \: Lebesgue \: measurable}, 0 \leq \varphi(x) \leq f(x) \: \mathrm{on} \: E, \: E_{\varphi} \subseteq E \: \mathrm{with} \: m(E_{\varphi}) < \infty, \: \varphi \: \mathrm{vanishes \: outside \: of \:} E_{\varphi} \right \}}$.

The definition above is rather complicated so we will take some time to explain it in detail.

First off, when we say that $\varphi$ vanishes outside of $E_{\varphi}$ we mean that for all $x \in E \setminus E_{\varphi}$ we have that:

(1)
\begin{align} \quad \varphi(x) = 0 \end{align}

In other words, for each $\varphi$ in the definition above we require there to be a subset $E_{\varphi} \subseteq E$ of finite measure for which $\varphi$ is nonzero on $E_{\varphi}$. We also require these functions $\varphi$ to be bounded and Lebesgue measurable. Then each $\varphi$ is bounded Lebesgue measurable and defined on a Lebesgue measurable set $E_{\varphi}$ with $m(E_{\varphi}) < \infty$, and so $\displaystyle{\int_{E_{\varphi}} \varphi}$ is well-defined.

The integral of $f$ on $E$ is then defined to be the supremum of all such integrals.

 Definition: Let $f$ be a nonnegative Lebesgue measurable function defined on a Lebesgue measurable set $E$. Then $f$ is said to be Lebesgue Integrable on $E$ if $\displaystyle{\int_E f < \infty}$.