My question is: For $f_n, g, h \in L^p(X)$, where $X$ is a finite measure space, if $f_n$ converges to $g$ a.e and $f_n$ converges to $h$ weakly in $L^p$, can we conclude any relationship between $g$ and $h$? Thanks!
Real Analysis Convergence question
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real-analysis
measure-theory
functional-analysis
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3Welcome to math.stackexchange! What does "$g$ converge to $h$" mean? Usually, we talk about convergence for a sequence (or a net), not for a single function. – 2012-01-25
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0Thanks, Davide! I have edited the question. – 2012-01-25
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1On which measured space do you work? – 2012-01-25
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1On any finite measure space. – 2012-01-25