= Solution
Let $N$ be the <vector> subspace of integrable <functions> that vanish <almost everywhere>. The <Lebesgue space> is the <quotient vector space> $L^1(\mathbb R)=\mathcal L^1(\mathbb R)/N$: its elements are equivalence classes of <functions> equal <almost everywhere>. Addition and scalar multiplication are $[f]+[g]=[f+g]$ and $c[f]=[cf]$.
Set $\|[f]\|_1=\int|f|$. This is well-defined because replacing $f$ by an equal-almost-everywhere representative does not change the <integral>. It is nonnegative, homogeneous, and obeys the triangle inequality by $|f+g|\leq|f|+|g|$. Part (i) says its only zero element is the zero class. Hence \b[it is a <norm> on $L^1$].
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