## Pointwise convergence implies other type of convergence

Let be a measure space for which . Let . Suppose that is a sequence in such that and exists for -a.e. . Prove that .

*PHD 4324 (Indiana)*

**Proof: **If we assume that then there exists a subsequence denoted also such that . Then is bounded in and is reflexive there exists a subsequence converges weakly to in …

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Categories: Measure Theory, Real Analysis
measure space

There is an other proof which uses Egoroff’s theorem.

Here are the arguments: let . We fix and such that uniformly on and . We have

,

so , which gives the result.

Thank you very much for your solution. 🙂