I'm learning about $L^p$ spaces and I'm having trouble with this exercise:
Let $\mu$ a measure in a set $X$, $p\geq1$ and
$L^p_w=\left\{f:X\to\mathbb{\overline{R}}\text{ such that $f$ is measurable and }\sup_{t\gt0}t^p\mu(\{x:|f(x)|>t\})<\infty\right\}.$ Show that if $\mu$ is finite, then $L^p_w \subset L^1$.
I know $f\in L^1 \iff \int_X|f(x)|d\mu(x)<\infty$, so I started by rewriting the integral as:
$\int_X|f(x)|d\mu(x)=\int_{\{x:|f(x)|\leq t_0\}}|f(x)|d\mu(x) + \int_{\{x:|f(x)|\gt t_0\}}|f(x)|d\mu(x)$
for some $t_0>0$.
Am I on the right track? Anyway, I was able to prove that the first summand is finite but I was having trouble with the second part. I'll appreciate any help.
P.S.: The previous problem was about to prove that $L^p\subset L^p_w$ and I did it, but I don't think it's useful for this new exercise...
This is false for $p=1$. A counter-example is $f(x)=\frac 1 x$ for $x>0$ and $0$ for $x \leq 0$ on the real line.
For $p>1$ use the following: $$\int |f(x)|d\mu (x) =\int_0^{\infty} \mu \{x: |f(x)| >t\} dt $$ $$=\int_0^{1} \mu \{x: |f(x)| >t\} dt+\int_1^{\infty} \mu \{x: |f(x)| >t\} dt$$ $$ \leq \mu (0,1)+\int_1^{\infty}\frac 1 {t^{p}}dt <\infty.$$