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If $f_k\in L^p(\mathbb{R}^n)\cap L^r(\mathbb{R}^n) $ for some $p,r\in [1,\infty), f_k\rightarrow g$ in $L^p(\mathbb{R}^n)$ and $f_k\rightarrow h$ in $L^r(\mathbb{R}^n)$, then $g=h$ $a.e.$ in $\mathbb{R}^n$

I took $1\leq p<r$ WLOG and put $p= (1-t)1+tr$ where $t\in [0,1)$ then I applied convexity property of function $|f_k-g|^p$ but its not going too far.

Shrey
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    One way (not the only way) is to use the fact that if a sequence converges in $L^p$ then a subsequence converges a.e. Do you see how to use this fact for this result? – Ian Feb 04 '16 at 17:33
  • I don't see it yet but I can think more about it. Is there a way to do it using convexity property of $a^x$ and then integrating. – Shrey Feb 04 '16 at 17:47
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    I don't think so. The problem is that you don't really have a pointwise definition of $g$ or $h$ from your hypotheses. That is, you know them as limits in $L^p$ and $L^r$ but that doesn't actually help you evaluate them anywhere (even off some null set). Using my suggestion gives you a formula for $g$ and a formula for $h$, which hold off some null set. – Ian Feb 04 '16 at 19:59

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