Show that any two probability measures which agree on a $\pi$-system also agree on the $\sigma$-algebra generated by that $\pi$-system.

State Fubini's theorem for non-negative measurable functions.

Let $\mu$ denote Lebesgue measure on $\mathbb{R}^{2}$. Fix $s \in[0,1)$. Set $c=\sqrt{1-s^{2}}$ and $\lambda=\sqrt{c}$. Consider the linear maps $f, g, h: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

$$f(x, y)=\left(\lambda^{-1} x, \lambda y\right), \quad g(x, y)=(x, s x+y), \quad h(x, y)=(x-s y, y)$$

Show that $\mu=\mu \circ f^{-1}$ and that $\mu=\mu \circ g^{-1}$. You must justify any assertion you make concerning the values taken by $\mu$.

Compute $r=f \circ h \circ g \circ f$. Deduce that $\mu$ is invariant under rotations.