Product Spaces and Integrals of Indicator Functions

Question

Let $(\mathscr{X}, \mathcal{A}, \mu)$ and $(\mathscr{Y},\mathcal{B}, \lambda)$ be two measure spaces (also assume that $\mu(\mathscr{X}), \lambda(\mathscr{Y})<\infty$) and $(\mathscr{X}\times\mathscr{Y},\mathcal{A}\otimes\mathcal{B},P)$ be the corresponding product space. Then for any $A\in \mathcal{A}$ and $B\in\mathcal{B}$ I would like to show that: \begin{equation*} \int_{\mathscr{Y}}1_{A\times B}d\lambda = 1_A\lambda(B) \end{equation*} Intuitively I get why this identity holds, but I am really struggling to formally prove it. Any help would be greatly appreciated! Thanks.

Answer 1

The function $1_{A\times B}(x, y)$ takes the value $1$ wherever $x\in A$ and $y\in B$; otherwise its value is $0$. If you are integrating the function over the second variable, just keep $x$ fixed.

In fact, note that you can express this function as a product of two single variable functions: $$1_{A\times B}(x, y) = 1_{A}(x)1_{B}(y).$$ As you are only integrating in the $y$ variable, this should make your computation easier.