Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence?

Let us also assume the measure space is $\sigma$-finite.

I'm asking because for continuous functions on the unit interval, weak convergence is equivalent to pointwise convergence.

  • 2
    $\begingroup$ Pointwise convergence alone does not imply weak convergence for continuous functions on the unit interval. You also need boundedness. $\endgroup$ – Daniel Fischer Feb 6 '16 at 15:59

It isn't true.

The standard counterexample is to look at $L^2((0,1))$ with Lebesgue measure and take $f_n(x) = \sqrt{2} \sin(n \pi x)$. The functions $f_n$ are orthonormal in $L^2$, so by Bessel's inequality they converge weakly to 0. But pointwise, the sequence $\{f_n(x)\}$ diverges for every $x \in (0,1)$.


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