Let $f:\mathbb{R}^n\to\mathbb{R}$ be a locally Lebesgue integrable function, ie $f$ is Lebesgue measurable and $$\int_{K}|f(x)|dx<+\infty$$ for every compact subset $K\subset\mathbb{R}^n$. Is it true that
$$\lim_{a\to 0^+}\int_{|x|\leq a}|f(x)|dx=0? \tag1\label1$$
I know that if $0\leq a_1<a_2$ then $$0\leq\int_{|x|\leq a_1}|f(x)|dx\leq \int_{|x|\leq a_2}|f(x)|dx,$$ but I really don't know how to proceed proving \eqref{1}.
Thanks in advance for any hint.
This is immediate from DCT. $I_{\{x: |x| \leq a\}} \leq I_{\{x: |x| \leq 1\}}$ whenever $|a| \leq 1$ and $fI_{\{x: |x| \leq 1\}}$ is integrable. Of course $fI_{\{x: |x| \leq a\}} \to fI_{\{0\}}$ as $a \to 0+$.