State Dynkin's $\pi$-system $/ d$-system lemma.

Let $\mu$ and $\nu$ be probability measures on a measurable space $(E, \mathcal{E})$. Let $\mathcal{A}$ be a $\pi$-system on $E$ generating $\mathcal{E}$. Suppose that $\mu(A)=\nu(A)$ for all $A \in \mathcal{A}$. Show that $\mu=\nu$.

What does it mean to say that a sequence of random variables is independent?

Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all uniformly distributed on $[0,1]$. Let $Y$ be another random variable, independent of $\left(X_{n}: n \in \mathbb{N}\right)$. Define random variables $Z_{n}$ in $[0,1]$ by $Z_{n}=\left(X_{n}+Y\right) \bmod 1$. What is the distribution of $Z_{1}$ ? Justify your answer.

Show that the sequence of random variables $\left(Z_{n}: n \in \mathbb{N}\right)$ is independent.