Absolute Value Criterion for Lebesgue Integrability of Measurable Functions

Recall from the Criterion for a Measurable Function to be Lebesgue Integrable page that we saw that if $f$ is a measurable function on an interval $I$ and if there exists a Lebesgue integrable function $g$ such that $\mid f(x) \mid \leq g(x)$ almost everywhere on $I$ then $f$ is Lebesgue integrable on $I$.

We will now use this theorem to make a nice characterization between the Lebesgue integrability of a function $f$ and its absolute value, $\mid f \mid$.

Theorem 1: Let $f$ be a measurable function on $I$. Then $f$ is Lebesgue integrable on $I$ if and only if $\mid f \mid$ is Lebesgue integrable on $I$.

• Proof: Let $f$ be a measurable function on $I$.

• $\Rightarrow$ Suppose that $f$ is Lebesgue integrable on $I$. Consider the positive and negative parts of $f$ denoted $f^+$ and $f^-$ and defined by:

(1)

\begin{align} \quad f^+ = \max \{ f, 0 \} \quad \mathrm{and} \quad f^- = \max \{ -f, 0 \} \end{align}

• Then $f^+$ and $f^-$ are both Lebesgue integrable on $I$ and we prove don the Lebesgue Integrability of the Positive and Negative Parts of a Function page. Noting that $\mid f \mid = f^+ + f^-$ we see that $\mid f \mid$ is Lebesgue integrable on $I$ (as we already proved on the Lebesgue Integrability of the Absolute Value of a Function page.

• $\Leftarrow$ Suppose that $\mid f \mid$ is Lebesgue integrable on $I$. Now we are given that $f$ is a measurable function, and so $f^+$ and $f^-$ are measurable functions. Moreover, it's not hard to see that the following inequalities holds almost everywhere on $I$:

(2)

\begin{align} \quad \mid f^+(x) \mid = f^+(x) \leq \mid f(x) \mid \quad \mathrm{and} \quad \mid f^-(x) \mid = f^-(x) \leq \mid f(x) \mid \end{align}

• But with these inequalities, we have by the theorem presented on the Criterion for a Measurable Function to be Lebesgue Integrable page that both $f^+$ and $f^-$ are Lebesgue integrable on $I$. Notice that:

(3)

\begin{align} \quad f = f^+ - f^- \end{align}

• So $f$ is the difference of two Lebesgue integrable functions and so by the Linearity of Lebesgue Integrals we have that $f$ is Lebesgue integrable on $I$. $\blacksquare$