Absolute continuity of Lebesgue integral for $L^2$ function

Question

Let $m$ be the Lebesgue measure on $\mathbb{R}$. If $f\in L^2(\mathbb{R})$, prove that for given $\epsilon>0$, there exists $\delta>0$ such that $\int_A |f|<\epsilon$ whenever $m(A)<\delta$.

I know the proof when $f$ is just integrable, but how should I proceed when $f$ is in $L^2$?

Any hints or advices are welcome! Thank you for your help!

Answer 1

By Cauchy-Schwarz, $$ \int_A|f|=\int_{\mathbb{R}}|f|1_A\leq ||f||_2m(A)^{\frac{1}{2}}$$ so you can take $\delta=\big(\frac{\varepsilon}{||f||_2+1}\big)^2$.