Friday, 11 September 2015

real analysis - the conditions for a measurable function to be the uniform limit of simple functions

In our homework we are asked to prove that, on a measurable space (Ω,F), every function f:ΩR,f0 can be written as the uniform limit of an increasing limit of simple functions.



However I looked up Real Analysis by Royden and baby Rudin, and here a boundedness condition is needed. Is it true? or the conclusion holds even without the boundedness condition?

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