0

Let $\{f_n\}_{n = 1}^{\infty}$ be a sequence of nonnegative measurable functions and let $f$ also be a nonnegative measurable function defined on some measure space $X$ with measure $\mu$. Is it true that if $$ \int_{E} f_n \, d\mu \to \int_{E} f \, d\mu $$ for every measurable set $E$, then $f_n \to f$ almost everywhere.

Epsilon-Delta
  • 147
  • 3
    Isn't https://math.stackexchange.com/questions/1412091/the-typewriter-sequence still a counterexample here. The convergence of integrals is certainly true as the $f=0$ – Jonas Lenz Jun 22 '24 at 10:53
  • Your hypothesis holds with just convergence in measure (instead of a.e. convergence) as long as the functions are dominated by an integrable fubction . In particular if the functions are unformly bounded and the measure is finite. So expecting a.e. convergence is too ambitious. – Kavi Rama Murthy Jun 22 '24 at 11:27

0 Answers0