Let $f_n : [0,1] \rightarrow \mathbb{R}$ be a sequence of monotone decreasing measurable functions $f_n \geq f_{n+1}$ that converges pointwise to $f: [0,1] \rightarrow \mathbb{R}$. What would be the counterexample which shows that Monotone convergence theorem does not apply for such sequence?
Besides original version of MCT (non-negative monotone increasing sequence) we also stated slightly different version where MCT applies also if ${f_n}$ are measurable, monotone increasing and $f_1$ is integrable.
My intuition is that I should come up with the sequence of functions that are measurable but not integrable (they need to converge though), however I did not manage to come up with something so far. Thanks for your help in advance!
Let $E_{k}=(1/2^{k+1},1/2^{k}]$, let $f_{n}$ be the functions defined in $[0,1]$ by $$ f_{n}(x) = \sum_{k=n}^{\infty} 2^{k+1}\chi_{E_{k}}(x), $$ where $\chi_{E_{k}}$ is the characteristic function of $E_{k}$. The sequence $\{f_{n}\}$ decreases pointwise to the zero function, but the integrals are all infinite.