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Short question: (Just an example. I want to know if similar thoughts can be used for other sequences of functions)

If I want to evaluate $\lim_{n\rightarrow \infty}\int_{[0,1]}-nxdx$, I can't do that with the monotone convergence theorem, since $f_n(x)=-nx$ isn't monotone increasing, but it is monotone decreasing.

But if I write instead $-\lim_{n\rightarrow \infty}\int_{[0,1]}nxdx$ the integrand is now monotone increasing. So I must be able to use the Monotone Convergence theorem here, right?

Can I use always this method for monotone decreasing sequences of functions?

Then why the Monotone Convergence theorem is only stated for increasing sequences of functions?

Duke
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1 Answers1

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No, you can't always do that for decreasing sequences. MCT applies to increasing sequences of positive functions. Hence a decreasing sequence of negative functions is ok. Decreasing sequence of positive (or neither) functions, no.

Consider $f_n = \chi_{[n,\infty)}$ on the line. Or if you want a finite measure, consider $\frac1t\chi_{(0,1/n)}$ on $[0,1]$.

David C. Ullrich
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