Alternate definition of Lebesgue integral

Question

I'm am essentially reasking this question Two definitions of Lebesgue integration but for a different approach.

The answer given is intuitive, however I wondering how exactly to show the equality by using Fubini's theorem.

Answer 1

$$\int_X f(x)\,d\mu=\int _X\int_0^\infty1_{t<f(x)}\,dt\,d\mu=\int_0^\infty\int_X 1_{t<f(x)}\,d\mu\,dt=\int_0^\infty \mu(\{x:f(x)>t\})\,dt$$ The first equality follows since $1_{t<f(x)}$ is $1$ in the interval $[0,f(x)]$, zero elsewhere, so its integral is $f(x)$. The second is of course Fubini-Tonelli, and the third follows from the usual definition of integration.

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