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?
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 $[0,1]$.