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I came across the following problem and do not know how to proceed:

Let $\{f_n\}$ be a sequence of integrable functions defined on an interval $[a,b]$. Then I have to prove that

"If $f_n(x) \longrightarrow 0$ almost everywhere (a.e.) and the $f_n$'s are uniformly bounded,then $\int_{a}^{b}f_n(x)dx \longrightarrow 0$".

Can someone point me in the right direction? Thanks in advance for your time.

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1 Answer 1

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Use bounded convergence theorem.

Let $f_n$ is a sequence of measurable functions, supported on the set $E$ of finite measure and $f_n \rightarrow f$ pointwise on $E$. If $f_n$'s are uniformly bounded, then $$ \lim \int f_n = \int f $$

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  • $\begingroup$ @Potato: since a.e. was mentioned in the question, i assumed familiarity with them. $\endgroup$
    – user62089
    Apr 1, 2013 at 5:36
  • $\begingroup$ Ah, indeed. I read the question too quickly. Nevermind me. $\endgroup$
    – Potato
    Apr 1, 2013 at 5:36

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