How do i prove this mixture of Lebesgue measurable functions is Lebesgue measurable?

Question

Let $C:\mathbb{R}^n\times\mathbb{R}\rightarrow \mathbb{R}^{n+1}:((x_1,\cdots,x_n),x_{n+1})\mapsto (x_1,\cdots,x_{n+1})$

Let $f:\mathbb{R}^{n+1}\rightarrow [0,\infty]$ be a Lebesgue measurable function.

Let $g:\mathbb{R}\rightarrow [0,\infty]$ be a Lebesgue measurable function.

Now define $F(x,y)=(f\circ C)(x,y)g(y)$, $\forall (x,y)\in \mathbb{R}^n\times\mathbb{R}$

How do i prove that $F\circ C^{-1}:\mathbb{R}^{n+1}\rightarrow [0,\infty]$ is Lebesgue measurable?

Answer 1

The product of measurable functions is measurable.