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I have a general question about integrals of sequence of functions. Suppose $f_n \rightarrow f$ pointwise. Can I automatically say that $\lim_{n\rightarrow \infty} f_n = \lim_{n\rightarrow \infty}inf (f_n) = \lim_{n\rightarrow \infty} sup (f_n) = f$.

What about integrals and more specifically Lebesgue integrals. can I say that $\lim_{n\rightarrow \infty} \int f_n$ exists? If yes, is this equal to $\lim_{n\rightarrow \infty}inf \int f_n$ = $\lim_{n\rightarrow \infty}sup \int f_n$ = $\lim_{n\rightarrow \infty} \int f_n$?

Or do I need to assume that $\int f_n \rightarrow \int f$ for the latter to hold?

Thanks for your help!!

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1 Answer 1

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You can construct a sequence a functions $f_n$ on $[0,1]$ that converges pointwise to the indicator function of the rationals in $[0,1]$, for example by placing an interval of width $1 / n2^k$ around each rational $q_k \in {\mathbb Q} \cap [0,1]$ and letting the value of the function be $1$ on these intervals, and $0$ elsewhere. It should be clear from this example that you can't get the equality of integrals to hold.

However your first statement is true, and follows from pointwise definitions of limits.

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