Completeness of Lebesgue spaces

Statement

For any measure space and $1\le p\le\infty$ , the space $L^p$ is a Banach space: every sequence Cauchy in its norm has a limit in that norm.

Summable-increment argument

From a Cauchy sequence choose a subsequence $f_{n_j}$ with $\|f_{n_{j+1}}-f_{n_j}\|_p\le2^{-j}$ . For $p<\infty$ , Minkowski and monotone convergence imply that $\sum_j|f_{n_{j+1}}-f_{n_j}|$ lies in $L^p$ , hence is finite almost everywhere. The telescoping series defines a limit, and its norm tail is at most $\sum_{j\ge m}2^{-j}$ . The Cauchy property brings the whole sequence to the same limit. For $p=\infty$ , remove the countable union of exceptional null sets; the increment series then converges uniformly there.

References

John K. Hunter, Measure Theory, Chapters 4, 5 and 7.