If $X$ and $Y$ are two measurable spaces. Is it true that $X \times Y$ (cartesian product) is measurable?
I can not understand how is the sigma-algebra of $X \times Y$ ?
Could someone helo me to undertands this, pls. Thanks for your time and help, everyone.
On
For product $\sigma$-algebras and product measures, see Chapter 5 of Measure, Integration & Real Analysis, which is available at http://measure.axler.net/.
Unforunately I don't have the reputation to comment. First of all, I'm assuming that your question is asking how to endow the space $X \times Y$ with a $\sigma$-algebra in a natural way from the $\sigma$-algebras $M_X$ and $M_Y$ on $X$ and $Y$ respectively.
Suppose that $\pi_X$ and $\pi_Y$ are the projection maps from $X \times Y$ to $X$ and $Y$. One way to define a $\sigma$-algebra on $X \times Y$ (which is called the produce $\sigma$-algebra) is to take the $\sigma$-algebra generated by the collection. $$\{\pi_X ^{-1}(E_X): E_X \in M_X \} \bigcup \{\pi_Y ^{-1}(E_Y): E_Y \in M_Y \}$$