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?
space of bounded measurable functions
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measure-theory
functional-analysis
banach-spaces
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0Yes. 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
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0Look here: http://math.stackexchange.com/questions/71121/space-of-bounded-continuous-functions-is-complete – 2012-04-03
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0(just replace continuous with measurable) – 2012-04-03
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0@tb: did you mean "Show that uniform limit (when exists) is a pointwise limit?" – 2012-04-03
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0@Ilya: No. See the answer below. – 2012-04-03