Prove measurability of the set of lines starting from measurable set.

Question

Let $A$ be a measurable set on X-axis. Then we have a point $B=(s,h)$, where $h$ is greater than 0. Let us define a set $F$, which consists of lines which start from every point, which belongs to $A$ and end on point B. How do I prove that this set of lines is measurable and its $\lambda_2(F)=\lambda_1(A)\frac{h}{2}$, where $\lambda$ is a Lebesgue measure and its index is the dimension. Intuitively it is clear, but I don't know techniques for proving it.

Answer 1

For fixed $y \in (0,h)$ consider $F^{y} \equiv \{x: (x,y)\in F\}$. It is easy to see from the definition of $F$ that $F^{y}=\{\frac {ys} h+(1-\frac y h)a: a\in A\}$. This set is a translate of $cA$ where $c=1-\frac y h$. Hence its measure is $(1-\frac y h) \lambda (A)$. Now Fubini's Theorem shows that $F$ is measurable and its measure is $\int_0^{h} (1-\frac y h) \lambda (A)dy=\lambda (A)\frac h 2$.