This topic is to show how to prove the following statement:
if converges strongly to some in , then up to a subsequence, converges almost everywhere to in .
The proof relies on the so-called Tchebyshev’s inequality. To this end, we first observe that converges strongly to in means
We now apply the Tchebyshev’s inequality, indeed, for each one has
The right hand side of the above inequality can be dominated by
which implies that
Thus converges to in measure. It turns out that up to a subsequence, converges to almost everywhere.