yesterday I asked a question and we get the answer, for reference this is what we ask
Problem 1 : Let $(X,M,\mu)$ be a measure space and $f$ is a real-valued function on $X$ such that $$\int_X |f| d\mu <\infty$$ . Then for any $\epsilon >0$ we can find a measurable set $E$ such that $\mu(E) <\infty$ and $$ \int_{X \backslash E} |f| d\mu <\epsilon.$$
There is another question which asked before,for reference this is the question :-
Probelm 2 : Find $\delta >0$ such that $\int_E |f| d\mu < \epsilon$ whenever $\mu(E)<\delta$
My question: Is there is a way can go from the first problem to the second one? which we should directly use the first problem in proving the second one.
As mentioned by David Mitra, the two problems are rather different. The first property states that $f$ is small enough, so that it takes small values on sets of infinite measure - a rough analogy on $\Bbb R^n$ would be that $f(x)\to 0$ as $x\to\infty$. In contrast, the second property says that the measure $\mathrm d\nu = f\;\mathrm d\mu$ is absolutely continuous w.r.t. $\mu$ which of course uses the fact that $f$ is not incredibly big but in essence the issues here may arise even if $\int f\mathrm d\mu$ is finite. Yet again, roughly it means that the first property is violated if $f = \infty$ on positive $\mu$ sets, whereas the second is violated if $f = \infty$ on null $\mu$ sets.
That said, in math one can rarely really safely claim that two results are completely unrelated.