Showing that a function is bounded in $L^1$ given a bound on its distribution function

Question

Let $f \in L^2((0,T)\times\Omega)$ where $\Omega$ is a compact manifold.

Suppose I know that for every $k > 0$, $$\mu(\{|f| > k\}) \leq Mk^{-\frac 12}$$ for some constant $M$ (which is independent of $k$).

Does it follow that $$\int_0^T\int_\Omega |f| \leq C(M)$$ for some constant $C$ that depends on $M$?

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I am wondering if some simple of seeing this if it is true at all. I am aware of the method using weak $L^p$ spaces (thanks to an answer on my previous thread) but that method is not easy to adapt.

Answer 1

Instead of $(0,T)\times\Omega$ work on $I=[0,1]$. Let \begin{align*} f_n(x)=\begin{cases} x^{-2} & x\in[1/n;1]\\ 0 & x\in [0;1/n] \end{cases} \end{align*} Then $|\{|f_n|>k\}|\leq |\{x^{-2}>k\}|=k^{-1/2}$ but $\lim_{n\to\infty}\int_0^1f_n(x)dx=\infty$.