Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 22H / a / Solution

If $f=g$ almost everywhere, then $|f|=|g|$ almost everywhere, so the integral defining $\Vert [f]{\Vert }_1$ is independent of the representative. Homogeneity and the triangle inequality follow from the corresponding pointwise properties of absolute value. Moreover,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 $1\le p\le \infty$. Applied with $p=1$, it says that every $\Vert \cdot {\Vert }_1$-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.