I am really stuck on this question. I am new in real analysis but interested. yet, I am not familiar with this.
Let $(X, \mathcal{A},\mu) = (Y, \mathcal{B}, v) = (\mathbb{N}, \mathcal{M}, c)$. Here, $\mathbb{N}$ is the set of natural numbers, $\mathcal{M} = 2^{\mathbb{N}}$, and $c$ is the counting measures defined by setting $c(E)$ equal to the number of points in $E$ if $E$ s finite and $\infty$ if $E$ is infinite set. Define $\displaystyle{f: \mathbb{N} \times \mathbb{N} \implies \mathbb{R}}$ by setting
$f(x,y) = 2-2^{-x}$ if $x = y$, or $f(x,y) = -2 + 2^{-x}$ if $x = y + 1$ or $f(x,y) = 0$ otherwise
Show that $f$ is measurable with respect to the product measure $c \times c$.
How to do this? I don't know how but can you help me through this?
Every function from $\mathbb N \times \mathbb N$ to $\mathbb R$ is measurable w.r.t. the product sigma algebra. Hence, it is measurable w.r.t. any measure on the product space. [ Measurable w.r.t. to a measure means it is measurable w.r.t the completion of the sigma algebra on $\mathbb N \times \mathbb N$ w.r.t. the measure, so if a function is already measurable w.r.t the product sigma algebra it is also measurble w.r.t any measure].