I ask because this seems to allow one to replace compactness with weak-sequential-compactness(and weak-compactness for Banach spaces of measurable functions) in some arguments. For example, if X is such a space of measurable functions then any weakly-sequentially-compact subset will have the property that convergence in measure is equivalent to convergence in X.
Maybe Proposition 1 of the paper "Theorems on representation of functions by series"
by K.S. Kazarian and D. Waterman can be helpful for some cases? You can download the paper from my page in RG.
I do not see how it helps, I am afraid. It is strong convergence in a regular F-space which implies convergence in measure. I would love to know, too, if there are spaces other than those where a ball consists of equicontinuous functions of a sense, such that weak sequential convergence implies convergence in measure.
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