Suppose $(X,\mathcal{B},\mu)$ is a measure space and for $y\in\mathbb{R}$ suppose that the function $f(\cdot,y):X\to X$ is measurable and $f(x,\cdot)$ is continuous. In many cases, I have seen that one can write that for a fixed $y$, the mapping $x\mapsto f(x,y)$ is $\mathcal{B}$-measurable. Is there some way that we can describe the mapping $(x,y)\mapsto f(x,y)$? The best way that I could phrase it is the following:
- For a fixed $y$, the mapping $x\mapsto f(x,y)$ is $\mathcal{B}$-measurable, and for a fixed $x$, the mapping $y\mapsto f(x,y)$ is $C^0$.
- $(x,y)\mapsto f(x,y)$$\mathcal{B}$-measurable $\times$$C^0$ (but this just looks wrong)
