Is $g(x):=F(f_1(x),f_2(x),...,f_n(x))$ measurable where F is continuous and each f is measurable. Let $f_1,..., f_n: \mathbb{R}\rightarrow\mathbb{R}$ be measurable functions. And $F:\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous. 
For $g:\mathbb{R}\rightarrow\mathbb{R}$, $g(x):=F(f_1(x),f_2(x),...,f_n(x))$. Is $g$ measurable?
I think I was be able to do that if F is measurable, but I don't know how is it if F is only continuous.

It is well known that there exist continuous functions $f : [0,2] \to \mathbb{R}$ that are not Lebesgue-measurable. Extend a function of this to the rest of the real line continuously, via 
$$
h(x) = \cases{f(0) \quad x < 0 \\ f(x) \quad 0 \leq x \leq 2 \\ f(2) \quad x > 2}
$$
Now, take $n = 1$, $f_1 \equiv id$, $F \equiv h$.
