= Solution
The <Lp space> is the vector space of equivalence classes, modulo equality almost everywhere, of measurable functions $f$ with $\int|f|^p\,d\mu<\infty$, equipped with $\|f\|_p=(\int|f|^p\,d\mu)^{1/p}$. To prove it is a <Banach space>, let $(f_n)$ be Cauchy in this norm. Choose a subsequence $(f_{n_j})$ such that
$$
\|f_{n_{j+1}}-f_{n_j}\|_p\leq2^{-j}.
$$
Take measurable representatives and put $g_j=|f_{n_{j+1}}-f_{n_j}|$, $G_M=\sum_{j=1}^Mg_j$. The <Minkowski inequality> gives $\|G_M\|_p\leq\sum_j2^{-j}\leq1$. By the <monotone convergence theorem>, $G=\sum_{j\geq1}g_j$ satisfies $\int G^p\,d\mu\leq1$. Consequently $G$ is finite almost everywhere, and the telescoping series
$$
f=f_{n_1}+\sum_{j\geq1}(f_{n_{j+1}}-f_{n_j})
$$
converges absolutely almost everywhere. Define it arbitrarily on the null exceptional set. The bound $|f|\leq|f_{n_1}|+G$ puts $f$ in $L^p$.
For the tails, monotone convergence and Minkowski again give
$$
\|f-f_{n_J}\|_p\leq\left\|\sum_{j\geq J}g_j\right\|_p\leq\sum_{j\geq J}2^{-j}\longrightarrow0.
$$
Since the original sequence is Cauchy, the <triangle inequality> then gives $f_n\to f$ in norm, not merely along the subsequence. Hence
$$
\boxed{L^p(E,\mathcal E,\mu)\text{ is complete for every }1\leq p<\infty.}
$$
This proof needs no finiteness or sigma-finiteness assumption on the measure space.
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