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I'm studying the various types of convergence for sequences of real valued functions defined on measure spaces: pointwise convergence a.e. , convergence in $L^p$ norm, weak convergence in $L^p$, and convergence in measure. It is quite complicated to see exactly what the relationships between the various types of convergence are, here is a diagram that shows the possibilities (when dealing with measurable subsets of $\mathbb{R}^N$, with $1 \leq p < \infty$ ; note that areas (2) and (3) are empty when the space has finite measure, since in this case convergence a.e. implies convergence in measure). enter image description here

Working mainly with these lecture notes (sorry, they're in Italian... anyway, many examples are on pages 124, 125), I have managed to fill all except the shaded areas in the diagram. So here are my questions:

Do there exist sequences of real valued funcions, defined on measurable subsets of $\mathbb{R}^N$ that:

  • $(6)$ converge in measure, but not a.e. and not weakly in $L^p$ for some $p \in [1, +\infty)$
  • $(7)$ converge in measure, weakly in $L^p$, but not a.e. and not in $L^p$
  • $(8)$ converge in measure, weakly in $L^p$, a.e., but not in $L^p$

I don't really know if I'm missing something stupid here, or if these examples are really tricky to find (or perhaps they don't exist).
Any help is much appreciated, thank you.

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1 Answer 1

up vote 6 down vote accepted

For (8), if $p>1$, let $a_n=\frac{1}{m(B(0,1/n))^{1/p}}$, and take $f_n=a_n\chi_{B(0,1/n)}$.

For (6) and (7), first let $A_1, A_2,A_3,\ldots$ be a sequence of subsets such that $m(A_n)\to 0$ and every element of the domain is in $A_n$ for infinitely many $n$.

For (6), let $(a_n)$ be a sequence of positive numbers such that $\displaystyle{\lim\limits_{n\to\infty}a_n m(A_n)^{1/p}=+\infty}$, and take $f_n=a_n\chi_{A_n}$.

For (7), if $p>1$, let $\displaystyle{b_n=\frac{1}{m(A_n)^{1/p}}}$, and take $f_n=b_n\chi_{A_n}$.

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Yes, thanks David. –  Jonas Meyer Mar 11 '12 at 3:40
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