I am reading something on Monotone convergence Theorem and I came across this statement "MCT Need not hold for decreasing sequence of functions",
My concern is not with the counterexample given under the statement but the language used here. Does this mean the same thing as " MCT doesn't always hold for decreasing sequence of functions"?
In which case I will take it to mean that, It can actually hold for some decreasing sequences of functions but not always so.
Because I was thinking about this counterexample, consider the sequence of mble functons $$f_n=\frac{1}{n}\chi_{[0,\frac{1}{n}]}$$
on $[0,1] $, clearly $f_n$ is a decreasing sequence
I could say $|f_n|\leq 1, \forall n$ and $f_n\to 0$ pointwise a.e, Hence I can apply Lebesgue Dominated convergence Theorem. $$\lim_{n\to \infty}\int f_n=\int 0$$
I'm not sure if it's also okay to say, redefining $$h_n=1-f_n$$ we get an increasing sequence with $h_n\to 1$ and this satisfies MCT even though $f_n$ is decreasing. and $$ \lim_{n\to \infty}\int h_n=\int 1$$
A statement that "Property A needs not hold for objects B" means that, there exists object B such that property A fails to hold. It does not mean that for all object B, property A does not hold.
So yes you can find an example of decreasing sequence that MCT does not hold, and you may also find an example of decreasing sequence that MCT actually holds.