Measurability of a two variable function

Question

I wonder if someone can help me solve this question (it's not homework):

Let $(X,\mathcal{A},\mu)$ be a measure space and let $f,h\in \mathcal{L}^1(X,\mathcal{A},\mu,\mathbb{C})$ be integrable functions.

Let $g: X \times X \to \mathbb{C}$ be defined by $g(x,y)=f(x)h(y)$.

Prove that $g$ is $\mathcal{A}\times\mathcal{A}$ measurable.

Answer 1

The product $\mu:\mathbb{C}\times \mathbb{C}\rightarrow \mathbb{C}$ defined by $\mu(x,y)=xy$ is measurable. The function $h:X\times X\rightarrow \mathbb{C}\times \mathbb{C}$ defined by $h(x)=(f(x),g(x))$ is measurable. This implies that $\mu\circ h$ is measurable.