transformation for my self

Question

I am trying to prove this equation of integral transformation with Borel measure. But so far have no idea. Can you help me how to start. Suppose $(X, A, \mu)$ is a $\sigma$-finite measure space and $f: X\to [0,\infty]$ an $A$-measurable function. Let $\nu$ be a Borel measure on $[0,\infty)$ such that $\nu(K)<\infty$ for any compact set $K$ in $[0,\infty)$. Show that

$$\int_X \nu([ 0 ,f(x))\;\mu(dx) = \int_0^\infty\mu(\{\ x\in X \ | f(x)>t\})\;\nu(dt).$$

Answer 1

This is often called the layer-cake representation. Compute

\begin{align*} \int_{X} \nu([0,f(x))\, \mu(dx) &= \int_{X} \int_{0}^{\infty} \mathbb{1}_{[0, f(x))}\, \nu(dt)\, \mu(dx) \\ &\overset{(i)}{=} \int_{0}^{\infty} \int_{X} \mathbb{1}_{[0,f(x))}\, \mu(dx)\, \nu(dt) \\ &\overset{(ii)}{=} \int_{0}^{\infty} \int_{X} \mathbb{1}_{\{x\, |\, f(x) > t \}}\, \mu(dx)\, \nu(dt) \\ &= \int_{0}^{\infty} \mu(\{x \, | \, f(x)>t \})\, \nu(dt). \end{align*}

Step $(i)$ follows from Tonelli's (Fubini-Tonelli) theorem (it applies as (a) $\mathbb{1}_{[0,f(x))}$ is measurable and non-negative and (b) $\mu \otimes \nu$ is $\sigma$-finite), while step $(ii)$ follows since $\mathbb{1}_{\{x\, |\, f(x) > t \}} = \mathbb{1}_{[0,f(x))}$.

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NOTE: Sometimes you may see the notation $\mathbb{1}_{L(f;t)(x)} := \mathbb{1}_{\{x\, |\, f(x) > t \}}$ instead (cf.Layer cake representation)

Let us know in the comments if there are any questions.

Answer 2

The question has been answered, however I'd like to ask how to prove, that $1_{[0,f(x)[} (t)$ is measurable since this is required for using Tonelli and therefore prove that given equality. We want to show that for every $C\in B[0,\infty]$ following holds: $(1_{[0,f(x)[})^{-1} (C) \in A $ x $B[0,\infty[$..but I'm struggling with this..