[Math] Show that $f$ is finite almost everywhere.

Question

Let $f:\mathbb{R} \rightarrow [0,\infty]$ be measurable so that $\int_{\mathbb{R}}^{}fdm<\infty$. Show that $f$ is finite almost everywhere.

Do I need to consider the subset $(0,1)\subset \mathbb{R}$ or by contradiction, let $E=\{x\in R,: |f|=\infty\}$, does it work?

Thanks in advance.

Answer 1

Can be seen from the following argument (essentially Chebychev's inequality for the $L^1$-norm). For every $\lambda>0$, \begin{eqnarray*} \infty>\|f\|_{L^1} & = & \int_{\{f>\lambda\}}f\; dm + \int_{\{f< \lambda\}}f\; dm \\ &\geq & \int_{\{f>\lambda\}}f\; dm \\ & \geq & \lambda\cdot m(\{f>\lambda\}). \end{eqnarray*} Therefore, for all $\lambda>0$, \begin{equation*} m(\{f>\lambda\}) \leq \frac{\|f\|_{L^1}}{\lambda}. \end{equation*}