Let $\Omega =[0,1] \times [0,1]$, $\mathcal{F} = \mathcal{M}_{[0,1]^2}$ (two-dimensional Lebesgue-measurable sets), $P=m_2$ (two-dimenional Lebesgue meaure). On the descibed probability space, consider the following two random variables:
$$X((\omega_1, \omega_2))=\omega_1$$ $$Y((\omega_1, \omega_2))=\omega_1*\omega_2$$
(a) Calculate the joint distribution of $(X, Y)$, does it have density?
(b) Describe the $\sigma$-field $\mathcal{F}_X$ generated by $X$ and calculate $\mathbb{E}(Y | \mathcal{F}_X)$ as a function of $\omega = (\omega_1, \omega_2)$ and also as a function of $X$.
For (a), the range is $0 \le X \le 1, 0 \le Y \le X$, then I would like to use the Jacobian to find the joint distribution $P_{(X,Y)}(B)$ for $B \in \mathcal{B}$ (Borel $\sigma$-field), then
$$P_{(X,Y)}(B)=\int_B \Biggl|\frac{1}{x}\Biggr| dm_2(x, y)$$ Maybe can anyone check is it correct?
For (b), I felt strange about the $\mathcal{F}_X$ generated by $X$, will it be $\mathcal{B}_{[0,1]}\times [0,1]$ ? because from $X=\omega_1$, we can only know about preimage of $\omega_1$, and random for $\omega_2$, but is it a $\sigma$-field ? Or did I make a mistake ? Hope for anyone's help.