Let $f:\mathbb{R}^n\to\mathbb{R}\cup\{-\infty,\infty\}$ be lebesgue integrable. Prove that $f$ is finite almost everywhere.
Attempt: Define \begin{equation}A=\{x\in\mathbb{R}^n:f(x)=\infty\}.\end{equation}
We need only prove that $\lambda_n(A)=0$, where $\lambda_n$ denotes Lebesgue measure on $\mathbb{R}^n$. Assume the contrary, that $\lambda_n(A)>0$. Since $f$ is Lebesgue integrable, we have the following bound:
\begin{equation} \left|\int_{\mathbb{R}^n}fd\lambda_n\right|<\infty. \end{equation}
Moreover, if $f$ is integrable we also have
\begin{equation} \int_{\mathbb{R}^n}|f|d\lambda_n<\infty. \end{equation}
But if $\lambda_n(A)>0$ then this inequality is not satisfied, which is a contradiction. So our result follows.
I am not sure how to justify rigorously the last step here - any help would be great. Is the remainder of the argument correct?