Show a measure is invariant under all translations by using compositions of reflections 
$f_x: (\mathbb{R}, \mathcal{B}) \to (\mathbb{R}, \mathcal{B})$ by
  $f_x(y)=2x-y$ and where $\mathcal{B}$ is the borel sigma-algebra
Let $\mu$ be a measure on $\mathcal{B}$ which is invariant under al
  reflections: $\mu(B)=\mu(f_x(B))$ for all $x\in \mathbb{R}$
  and $B\in\mathcal{B}$.  Furthermore suppose that $\mu([0,1])=k <
 \infty$.  Show, by using compositions of reflections, that $\mu$ is
  invariant under all translations and deduce that $\mu=k\lambda$ where
  $\lambda$ is Lebesgue measure.

I have really no idea how I can do this. First, I don't see how I can use reflections to show something about all translations. 
I think it need to be something like: $\mu$(compositions of reflections of a set E) = $\mu(E)$. But I don't know which compositions of reflections I need to use and how it is possible to prove something about all translations with that.

It is a basic fact that any translation is a composition of two reflections. Let's see what happens when we compose two reflections, say $f_x$ and $f_y$:
$$
f_x(f_y(a)) = f_x(2y - a) = 2x - (2y - a) = a + (2x - 2y).
$$
Voilà -- a translation by $2x - 2y$. By choosing $x$ and $y$ appropriately, we can get any translation we want.
That means if the measure is invariant under reflection, it is also invariant under translation. The hypothesis that $\mu([0,1]) = k < \infty$ is only necessary for the latter part, to show that $\mu = k \lambda$.
