I was studying about Lebesgue spaces and ran into the definition of essential supremum.
Actually I have seen two very similar definitions: Let $(X,\mathfrak{M},\mu)$ be a measure space and $f:X\to [-\infty,+\infty]$ be a measurable function. Then $$\text{ess sup}|f(x)|:=\inf\{c\in \mathbb{R}: \mu(\{x\in X:|f(x)|>C\})=0\}. \qquad(*)$$ Also you can find exactly the same definition where infimum is taken over $c>0$, i.e. $$\text{ess sup}|f(x)|:=\inf\{c>0: \mu(\{x\in X:|f(x)|>C\})=0\}. \qquad (**)$$
And I think that probably $(*)=(**)$.
It follows easily that $(**)\geq (*)$. But how to show the converse ineqaulity?
Can anyone provide the rigorous proof, please?