Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ ..

Question

Let $(X, M, \mu )$ be a space with measure. $f:X \to \mathbb R \text{ and } f\in L^1(X).$ Prove that for all $\epsilon > 0$ that there exists $\delta > 0$ such that for $E \in M$, $\mu(E)< \delta$ the following is true: $$|\int_E f d\mu|< \epsilon.$$ $(M)- \text{ is the sigma algebra assigned to this space.}$

This seems intuitively close to the notion of continuity. I know that if $f$ is a measurable function that $$\lambda(E)=\int_{E}fd\mu$$. $f$ is continuous, therefore measurable, I feel like I can use this, but don't know how.

Answer 1

Define $E_n = \{ x : |f(x)| \geq n \}$. Since $f \in L^1$, $\mu(\{ x : |f(x)| = \infty \} ) = 0$. Hence, by the dominated convergence theorem, $$ \lim_{n \rightarrow \infty} \int_{E_n} |f| d\mu = 0. $$ Let $\epsilon > 0$. Then there exists $N \geq 1$ such that $$ \int_{E_N} |f| d\mu < \epsilon/2. $$ Choose $\delta = \epsilon/(2N)$. Then if $E \in M$ and $\mu(E) < \delta$, we have \begin{align*} \int_E |f| d\mu &= \int_{E \cap E_N} |f| d\mu + \int_{E \cap E_N^c} |f| d\mu \\ &\leq \int_{E_N} |f| d\mu + \int_{E \cap E_N^c} N d\mu \\ &< \epsilon/2 + N \mu(E \cap E_N^c) \\ &\leq \epsilon/2 + N \delta \\ &< \epsilon/2 + \epsilon/2 = \epsilon. \end{align*}