The Lebesgue Integral for Lebesgue Measurable Functions

So far we have defined the Lebesgue integral for the following collections of functions:

• 1) Simple Functions: If $\varphi$ is a simple function defined on a Lebesgue measurable set $E$ with $m(E) < \infty$ and with canonical representation $\displaystyle{\varphi(x) = \sum_{k=1}^{n} a_k \chi_{E_k}(x)}$ then the Lebesgue integral of $\varphi$ on $E$ is:

(1)

\begin{align} \quad \int_E \varphi = \sum_{k=1}^{n} a_k m(E_k) \end{align}

• 2) Bounded Lebesgue Measurable Functions: If $f$ is a bounded Lebesgue measurable function defined on a Lebesgue measurable set $E$ with $m(E) < \infty$ then $f$ is said to be Lebesgue integrable on $E$ if $\displaystyle{\underline{\int_E f} = \overline{\int_E f}}$ and if we denote this common value by $\displaystyle{\int_E f}$ to be the Lebesgue integral of $f$ on $E$.

• 3) Nonnegative Lebesgue Measurable Functions: If $f$ is a nonnegative Lebesgue measurable function defined on a Lebesgue measurable set $E$ then the Lebesgue integral of $f$ on $E$ is:'

(2)

\begin{align} \quad \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 \} \end{align}

• We say that $f$ is Lebesgue integrable on $E$ if $\displaystyle{\int_E f < \infty f}$.

We can now define a more general version of a Lebesgue integral for Lebesgue measurable functions.

Definition: Let $f$ be a Lebesgue measurable function defined on a Lebesgue measurable set $E$. The Lebesgue Integral of $f$ on $E$ is defined as $\displaystyle{\int_E f = \int_E f^+ - \int_E f^-}$.

Recall that $f^+ = \max \{ f, 0 \}$ and $f^- = \max \{ -f, 0 \}$. Note that in both cases $f^+$ and $f^-$ are nonnegative Lebesgue measurable functions defined on $E$. So the general Lebesgue integral $\displaystyle{\int_E f}$ is defined in terms of the difference of two Lebesgue integrals for nonnegative Lebesgue measurable functions which we have already defined.

Definition: Let $f$ be a Lebesgue measurable function defined on a Lebesgue measurable set $E$. Then $f$ is said to be Lebesgue Integrable on $E$ if $|f|$ is Lebesgue integrable on $E$.

Proposition 1: Let $f$ be a Lebesgue measurable function defined on a Lebesgue measurable set $E$. Then $f$ is Lebesgue integrable on $E$ if and only if $\displaystyle{\int_E f^+ < \infty}$ and $\displaystyle{\int_E f^- < \infty}$.

• Proof: $\Rightarrow$ Suppose that $f$ is Lebesgue integrable on $E$. Then by definition $|f|$ is Lebesgue integrable on $E$. But $|f|$ is a nonnegative Lebesgue measurable function defined on $E$, so $|f|$ being Lebesgue integrable implies that:

(3)

\begin{align} \quad \int_E |f| < \infty \end{align}

• However $|f| = f^+ + f^-$. So by the additivity of the Lebesgue integral for nonnegative Lebesgue measurable functions we have that:

(4)

\begin{align} \quad \int_E f^+ + \int_E f^- = \int_E |f| < \infty \end{align}

• Therefore $\displaystyle{\int_E f^+ < \infty}$ and $\displaystyle{\int_E f^- < \infty}$.

• $\Leftarrow$ Suppose that $\displaystyle{\int_E f^+ < \infty}$ and $\displaystyle{\int_E f^- < \infty}$. Then by the additivity of the Lebesgue integral for nonnegative Lebesgue measurable functions we have that:

(5)

\begin{align} \quad \int_E f^+ + \int_E f^- = \int_E |f| < \infty \end{align}

• So $f$ is Lebesgue integrable on $E$. $\blacksquare$

Note that in general, $\displaystyle{\int_E |f| < \infty}$ by itself does NOT imply that $f$ is Lebesgue integrable. We also require that $f$ is Lebesgue measurable.