Let $ (X,\mathcal{S},\mu)$ be a finite measure space, let $f$ be $\mathcal{S}$-measurable and let $E_{n}:= \{x\in X :n-1\le |f(x)|<n\}$ for $n=1,2,\dots$
Show that:
$$f \in L_1\iff\sum_{n=1}^{\infty}n\mu(E_n)<+\infty$$
More generally for $1 \le p <\infty$ show that:
$$f\in L_p\iff\sum_{n=1}^{\infty}n^p\mu(E_n)<+\infty$$
Here is what I got by reading an intempt to prove the statement:
$$\chi_{E_1} + \frac{1}{2} \sum_{n=2}^{\infty}n\chi_{E_n}\le \sum_{n=1}^{\infty}(n-1)\chi_{E_n}\le|f|\le \sum_{n=1}^{\infty}n\chi_{E_n}$$
And integrating with power $p \in (1,\infty]$ we get:
$$2^{-p}\sum_{n=2}^{\infty}n^p\mu(E_n)+\mu(E_1)\le\int |f|^p\,d\mu\le\sum_{n=1}^{\infty}n^p\mu(E_n)$$
I need help to understand at all the inequalities above , also to write down a formal and detailed proof of the statement . Thanks so much
These inequalities do not seem to be the best ones to me. Why don't you try this: Note that the $E_n$ simply divide the domain of $E_n$ according to the integer part of $|f|$. In $E_n$, $f$ is "sandwiched" between $n-1$ and $n$. Writing this with functions: $$(n-1)\chi_{E_n}\leq|f|\chi_{E_n}\leq n\chi_{E_n}.$$ We can take the power $p$ on the inequalities above. Since characteristic functions only take values $1$ or $0$, then $$(n-1)^p\chi_{E_n}\leq|f|^p\chi_{E_n}\leq n^p\chi_{E_n}.$$ The left-side is not really what you want. But you will believe me if I say that, for $n$ sufficiently large (with "large" depending solely on $p$), $(n-1)^p\geq n^p-1$, or equivalently $n^p\left(\left(\frac{n-1}{n}\right)^p-1\right)\geq -1$. Then we have $$(n^p-1)\chi_{E_n}\leq |f|^p\chi_{E_n}\leq n^p\chi_{E_n}$$ Sum all these and integrate. What do you get?