= Solution
If $f=g$ almost everywhere, then $|f|=|g|$ almost everywhere, so the integral defining $\|[f]\|_1$ is independent of the representative. Homogeneity and the triangle inequality follow from the corresponding pointwise properties of absolute value. Moreover,
$$
\|[f]\|_1=0
\quad\Longleftrightarrow\quad f=0\text{ almost everywhere}
\quad\Longleftrightarrow\quad [f]=[0],
$$
so this is a norm on the <Lebesgue space> of almost-everywhere equivalence classes.
The <Riesz-Fischer theorem> states that $L^p$ is complete for $1\leq p\leq\infty$. Applied with $p=1$, it says that every $\|\cdot\|_1$-Cauchy sequence of these equivalence classes converges in $\mathcal L^1(\mathbb R^n)$. Thus $\mathcal L^1(\mathbb R^n)$ is a Banach space.
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