I have the following exercise to do, and I am totally lost with it:
Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X : \Omega → \mathbb{R}_{\ge0}$ a random variable with (cumulative) distribution function $F_X : \mathbb{R}_{\ge0} \to [0, 1]$. Further let $U_0 := X^{-1} (\{0\})$ and $U_n := X^{-1} ((n − 1, n])$ for $n \in \mathbb{N}$, so that $\{U_{n}\}_{n\ge0}$ forms a partition of $\Omega$. Construct the conditional expectation $E[X\mid\mathcal{H}]$ of $X$ with respect to the sub-$\sigma$-algebra $\mathcal{H} \subset \mathcal{F}$ generated by $\{U_n\}$ by the following steps:
a) Evaluate the measure $Q(H) :=\int_{H} X dP$ on $(\Omega,\mathcal{H})$ for the sets $U_n \in \mathcal{H}$. Express your result as an integral with respect to $F_X$.
b) For $n$ fixed, conclude that $E[X\mid\mathcal{H}]$ is constant on $U_n$. Determine $E[X\mid\mathcal{H}](\omega)$ for all $\omega\in\Omega$.
The second part seems clear, but I am totally lost with the first part, I would totally appreciate any tips.
$\int_{U_n} X\, dP=\int_{(n-1,n]} x \, dF_X(x)$. [In general $\int \phi (X) \, dP=\int \phi(x)dF_X(x)$. Put $\phi (x)= xI_{(n-1,n]}$]. $E(X|\mathcal H)=\sum c_n I_{U_n}$ where $c_n$ is computed using the equation $\int_{U_n} X\, dP=c_nP(U_n)$. Hence $c_n=\frac {\int_{(n-1,n]} x \, dF_X(x)} {P(U_n)}$.