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Can anyone give an example on (0,1) where the Lebesgue monotone convergence theorem is not true for decreasing sequences of non-negative functions? I can't find such an example, does anyone know? I would be grateful..

precelina m
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  • So is it a theorem, or not true? You can't have both, unfortunately. –  Dec 11 '20 at 20:19
  • This may be of interest. – David Mitra Dec 11 '20 at 20:33
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    This clearly does not exist: Your sequence (from a certain index on) is bounded by its first integrable member. Since the functions are all nonnegative, use Lebesgue's dominated convergence theorem. – Hyperbolic PDE friend Dec 11 '20 at 20:41
  • At the lecture, we defined it as a theorem, but as a task we had to find a sequence of non-negative functions for which it does not work.. – precelina m Dec 11 '20 at 21:59
  • The Lebegue Monotone Convergence theorem does not require the functions to have finite integral. So we cannot say that: The "sequence (from a certain index on) is bounded by its first integrable member". There might be the case that no member is integrable (have finite integral) and then we can NOT apply Lebesgue Dominate Convergence theorem. – Ramiro Dec 12 '20 at 14:53
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The Lebegue Monotone Convergence theorem does not require the functions to have finite integral. But it requires the functions to be a non-decreasing sequence.

Here is the counter-example you are looking for:

Consider the function $f_n: (0,1) \to\mathbb{R}$, defined as $f_n(x)=\frac{1}{nx}$. Note that $\{f_n\}_n$ is a decreasing sequence of non-negative functions and $f_n \to 0$ pointwisely, but, for all $n$, $\int_{(0,1)} f_n = +\infty $ does not converge to $0$.

Here is another counter-example:

Consider the function $f_n: \mathbb{R} \to\mathbb{R}$, defined as $f_n=\chi_{[n,+\infty)}$. Note that $\{f_n\}_n$ is a decreasing sequence of non-negative functions and $f_n \to 0$ pointwisely, but, for all $n$, $\int_{\mathbb{R}} f_n = +\infty $ does not converge to $0$.

Ramiro
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