the well-known Lebesgue’s dominated convergence theorem states that pointwise convergence of a sequence of functions implies convergence of the sequence of integrals if an integrable function dominating the sequence of functions almost everywhere can be found. My question: is the existence of such a dominating function also a necessary condition for the convergence of integrals? Or can one think of an example where the sequence of integrals does converge to the expected limit, but no dominating function can be found for the sequence of functions?
1
-
$\begingroup$ You should include: functions with values in $[0,\infty)$ - otherwise, there are obvious examples. $\endgroup$– Damian RösslerJul 25, 2013 at 14:28
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The existence of a dominating function isn't necessary.
Consider for instance the sequence of non-negative $L^1(0,1)$ functions defined by $$f_n(x)=(x-n^{-\beta})^{-\alpha} \mathrm{\ if\ } x>1/n^\beta, \mathrm{\ zero\ otherwise},\; n\geq 2,$$ where $0<\alpha<1$ and $\beta>0$.
It converges almost everywhere and in $L^1(0,1)$ to $f(x)=1/x^\alpha$, but the least dominating function has integral $$(1-\alpha)^{-1}\sum_{n\geq 1} (n^{-\beta}-(n+1)^{-\beta})^{1-\alpha},$$ which is infinite as soon as $(1-\alpha)(1+\beta)\leq 1$, e.g. for $\alpha=1/2$, $\beta=1$.
The right notion is that of uniform integrability (also called equi-integrability), used in Vitali convergence theorem.
