I would like some help in trying to solve this problem
We have to prove that if $ f \in L_p$ and $E_n$ = { $ x \in X : \vert f(x) \vert \geq n $}, then $\mu(E_n) \to 0 $ as $ n \to \infty$
My work:
We have that $\int_{X} \vert f \vert^p d\mu \lt \infty$. Then, since $E_n \subset X $ for each $n$, :
$\int_{E_n} \vert f \vert^p d\mu$ < $\infty$ for each $n$.
Also, for a natural $k$ we have that $\int_{E_k} k \space d\mu$ = $ k*\mu(E_k)$ $\leq$ $\int_{E_k} \vert f \vert^p d\mu$
Letting go $k \to \infty$, we have that $ k*\mu(E_k)$ $\lt \infty $ iff $\mu(E_k) \to 0$ (By convention $0 * \infty= 0$)
I want to know if my solution is correct or if its nonsense.
Looks good, but you can rearrange the equation to obtain something more recognizable.
$$ k^p\mu(E_k)\leq\int_{E_k}\vert f\vert^pd\mu\leq \int_X\vert f\vert^pd\mu $$
and
$$ \mu(E_k)\leq\biggl(\int_X \vert f\vert^pd\mu\biggr)k^{-p}\to 0 $$
by the Archimedean Property.