= Solution
Use <Lebesgue measure> restricted to the <Lebesgue measurable set> $E$. For $1\leq p<\infty$, start with the measurable real-valued <functions> $f$ on $E$ satisfying $\int_E|f|^p\,dm<\infty$. Define an <equivalence relation> by $f\sim g$ when $f=g$ <almost everywhere>, and let $L^p(E)$ consist of their <equivalence classes>. Addition and scalar multiplication are defined using representatives. The <Lp norm> is
$$
\boxed{\|[f]\|_p=\left(\int_E|f|^p\,dm\right)^{1/p},\qquad1\leq p<\infty.}
$$
For $p=\infty$, use measurable <functions> bounded <almost everywhere>, apply the same identification, and define the <L-infinity norm> by
$$
\boxed{\|[f]\|_\infty=\operatorname*{ess\,sup}_{x\in E}|f(x)|
=\inf\{C\geq0:m(\{x\in E:|f(x)|>C\})=0\}.}
$$
These expressions are independent of the representative. For finite $p$, zero <norm> is equivalent to $f=0$ <almost everywhere>: each set $\{|f|>1/n\}$ must have measure zero, and their union is $\{f\ne0\}$. The same implication follows from the <essential supremum> for $p=\infty$. Absolute homogeneity is immediate; <Minkowski inequality> proves the <triangle inequality> and closure under addition. Thus these are <normed vector spaces>. Using complex-valued representatives gives their complex versions by the same construction.
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