Question: Can we give an example to show that the measurability of the set $f^{-1}(\{a\})$ for every a does not imply that $f$ is a measurable function?

Answer: 
Since the geometric mean for both $(2,2)$ and $(1,4)$ is $2$, while the arithmetic means are $2$ and $2.5$, the answer is a clear no. The only thing you can say is that the geometric mean is smaller or equal to the arithmetic.
No it is not. Arithmetic mean gives you one equation. And there are two numbers to solve for. So there are infinite possibilities. For example, $$a=1,b=100$$ $$ a=0,b=101$$
They both have same A.M but widely different G.M