The statement for the continuity of integration is given below:
The statement for the continuity of measure is given below:
My question is:
Why in the case of continuity of measure we wanted $m(B_{1}) < \infty$ but we did not want this in case of continuity of integration? is it because we are allowing by footnote 7 on pg.79 $E_{n}'s$ to have infinite measure?
This is because $f$ is integrable on each $E_n$ with finite integral, since $|\int_{E_n}f| \leq\int_{E_n}|f| \leq \int_E|f|<\infty ,\forall n \in \Bbb{N}$ so $|\int_{E_1}f| \leq \int_E|f|<\infty$.
For the measures we must have $m(B_1)<\infty$.
Take for instance the decreasing sets $B_n=[n,+\infty)$ where $m(B_1)=+\infty$ and see what happens.