5
$\begingroup$

Let $(\Omega, \Sigma)$ be a measurable space. Is the space of bounded measurable functions $B_b(\Sigma)$ equipped with the supremum norm a Banach space, i.e. complete?

  • 0
    Yes. First: show that the pointwise limit of a Cauchy sequence exists, second: show that the pointwise limit is bounded, third: show that the pointwise limit is measurable. Fourth: show that the pointwise limit is a uniform limit. Which part is causing trouble?2012-04-03
  • 0
    Look here: http://math.stackexchange.com/questions/71121/space-of-bounded-continuous-functions-is-complete2012-04-03
  • 0
    (just replace continuous with measurable)2012-04-03
  • 0
    @tb: did you mean "Show that uniform limit (when exists) is a pointwise limit?"2012-04-03
  • 0
    @Ilya: No. See the answer below.2012-04-03

1 Answers 1