integral of sum of a sequence is sum of the integrals

[[concept]]

Topics

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Theorem

Let $\{f_n{\}}_n$ be a sequence in $L^{+}(E)$. Then ${\int }_E{\sum }_n{f}_n={\sum }_n{\int }_E{f}_n$

Proof

via induction. Since the lebesgue integral of sum is sum of the integral, we know for each $N\in \mathbb{N}$ that ${\int }_E{\sum }_{n=1}^N{f}_n={\sum }_{n=1}^N{\int }_E{f}_n$ And since ${\sum }_{n=1}^1{f}_n\le {\sum }_{n=1}^2{f}_n\le \dots$ and ${\sum }_{n=1}^N{f}_n\to {\sum }_{n=1}^{\infty }{f}_n$ as $N\to \infty$, by the Monotone Convergence Theorem we have

\int _{E} \sum_{n=1}^\infty &= \lim_{ N \to \infty } \int _{E} \sum_{n=1}^N f_{n} \\ &= \lim_{ N \to \infty } \sum_{n=1}^N \int _{E} f_{n} \\ &= \sum_{n=1}^\infty \int _{E} f_{n} \end{align}$$ $$\tag*{$\blacksquare$}$$

NOTE

this does not hold for Riemann integration. Enumerate the rationals and let $f_n$ be the function that is 1 for the first $n$ rational numbers and 0 everywhere else

References

References

See Also

Mentions

Mentions

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