(a) Let $\left(X_{n}\right)_{n \geqslant 1}$ and $X$ be real random variables with finite second moment on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that $X_{n}$ converges to $X$ almost surely. Show that the following assertions are equivalent:

(i) $X_{n} \rightarrow X$ in $\mathbf{L}^{2}$ as $n \rightarrow \infty$

(ii) $\mathbb{E}\left(X_{n}^{2}\right) \rightarrow \mathbb{E}\left(X^{2}\right)$ as $n \rightarrow \infty$.

(b) Suppose now that $\Omega=(0,1), \mathcal{F}$ is the Borel $\sigma$-algebra of $(0,1)$ and $\mathbb{P}$ is Lebesgue measure. Given a Borel probability measure $\mu$ on $\mathbb{R}$ we set

$$X_{\mu}(\omega)=\inf \left\{x \in \mathbb{R} \mid F_{\mu}(x) \geqslant \omega\right\}$$

where $F_{\mu}(x):=\mu((-\infty, x])$ is the distribution function of $\mu$ and $\omega \in \Omega$.

(i) Show that $X_{\mu}$ is a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ with law $\mu$.

(ii) Let $\left(\mu_{n}\right)_{n \geqslant 1}$ and $\nu$ be Borel probability measures on $\mathbb{R}$ with finite second moments. Show that

$$\mathbb{E}\left(\left(X_{\mu_{n}}-X_{\nu}\right)^{2}\right) \rightarrow 0 \text { as } n \rightarrow \infty$$

if and only if $\mu_{n}$ converges weakly to $\nu$ and $\int x^{2} d \mu_{n}(x)$ converges to $\int x^{2} d \nu(x)$ as $n \rightarrow \infty$

[You may use any theorem proven in lectures as long as it is clearly stated. Furthermore, you may use without proof the fact that $\mu_{n}$ converges weakly to $\nu$ as $n \rightarrow \infty$ if and only if $X_{\mu_{n}}$ converges to $X_{\nu}$ almost surely.]