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The idea is to prove completeness of $L^1$ using Tikhonov theorem. The proof will be for narrower class of functions though.

There is a Tikhonov theorem that states that for any set of compact topological spaces $\{M_{\alpha}\}$ the product $\prod_{\alpha}M_{\alpha}$ is also compact.

From this (with some restrictions) it follows that a Cauchy sequence of measurable functions $\{f_{n}\}$ is compact in the topology of pointwise convergence.

And all that is left is to prove that from pointwise convergence follows convergence in $L^{1}$. This fact can be found in any functional analysis course.

The question is what are the required conditions on each $f_{n}$ from the sequence. It should be bounded for sure but is it enough?

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