# How does one compute conditional expectation with respect to a continuous random variable

conditional probabilityconditional-expectationmeasure-theoryprobability theoryrandom variables

I can't wrap my head around the way to compute conditional expectation with respect to a continuous random variable. For instance, consider a probability space $$(\Omega, A, P)$$, where $$\Omega = [0,1]$$ and $$A$$ is a corresponding Borel $$\sigma$$-algebra. Let's define random variables $$X(\omega)=\omega$$ and $$Y=\sin (\pi \omega)$$. How does one compute the expression for $$\mathbb{E}(X\mid \mathcal{B})$$, where $$\mathcal{B}$$ is the $$\sigma$$-algebra generated by $$Y$$?

If $$Y$$ was discrete, such as
$$Y = \begin{cases} 1,&\omega\in[0,1/2]\\ 0,&\omega\in(1/2,1] \end{cases} ,$$

then I understand that $$\mathbb{E}(X\mid \mathcal{B})$$ would be equal to
$$\mathbb{E}(X\mid \mathcal{B}) = \frac{1}{4}\mathbb{I}_{Y=1}+\frac{1}{4}\mathbb{I}_{Y=0}.$$

However, in the continuous case I'm not so sure. My intuition suggests that it's something like $$\sin^{-1}(y)/\pi$$; however, then
$$\mathbb{E}(\mathbb{E}(X\mid Y)) = \int_Y \mathbb{E}(X\mid Y)d\omega = \frac{2}{\pi}\cdot\int_0^1 \arcsin(y) dy \neq \mathbb{E}(X).$$

For instance, consider a probability space $$(Ω,\mathcal A,P)$$, where $$Ω=[0,1]$$ and $$A$$ is a corresponding Borel σ-algebra. Let's define random variables $$X(ω)=ω$$ and $$Y=sin(πω)$$. How does one compute the expression for $$E(X∣\mathcal B)$$, where $$\mathcal B$$ is the σ-algebra generated by Y?
Using $$\arcsin: [0..1]\mapsto [0..\pi/2]$$
A null events described by $$\{\omega\in\Omega:Y(\omega)=y\}$$, (where $$y\in[0..1)$$ ) will contain exactly two outcomes with no bias: $$\{\omega\in\Omega:Y(\omega)=y\}=\{\arcsin(y)/\pi, 1-\arcsin(y)/\pi\}$$ Therefore $$\forall y\in Y(\Omega)~,\mathsf E(X\mid Y=y)=1/2$$, so since this holds for any $$y$$, ...$$\mathsf E(X\mid\mathcal B)=(1/2)\mathbf 1_{Y\in[0..1]}$$