Suppose $(X, \mathcal{M}, \mu)$ is a $\sigma$-finite measure space and $f \in L^{+}(X)$ where $f: X\to [0,\infty]$ are all measurable functions. Let $$G_f = \{ (x,y) \in X \times [0, \infty]: y \le f(x) \}.$$
The book gives the hint that the map $(x,y) \to f(x)-y$ is the composition of $(x,y) \to (f(x),y)$ and $(z,y) \to z -y$.
(1)My question is how to prove the map $h: (x,y)\to (f(x),y)$ is measurable by the coordinate function is measurable?
(2)And why $(z,y)=z-y$ is measurable by the continuous function?
(3)Also, what does the "$\sigma$-finite " work?