(a) Let $\left(X_{i}, \mathcal{A}_{i}\right)$ for $i=1,2$ be two measurable spaces. Define the product $\sigma$-algebra $\mathcal{A}_{1} \otimes \mathcal{A}_{2}$ on the Cartesian product $X_{1} \times X_{2}$. Given a probability measure $\mu_{i}$ on $\left(X_{i}, \mathcal{A}_{i}\right)$ for each $i=1,2$, define the product measure $\mu_{1} \otimes \mu_{2}$. Assuming the existence of a product measure, explain why it is unique. [You may use standard results from the course if clearly stated.]

(b) Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which the real random variables $U$ and $V$ are defined. Explain what is meant when one says that $U$ has law $\mu$. On what measurable space is the measure $\mu$ defined? Explain what it means for $U$ and $V$ to be independent random variables.

(c) Now let $X=\left[-\frac{1}{2}, \frac{1}{2}\right]$, let $\mathcal{A}$ be its Borel $\sigma$-algebra and let $\mu$ be Lebesgue measure. Give an example of a measure $\eta$ on the product $(X \times X, \mathcal{A} \otimes \mathcal{A})$ such that $\eta(X \times A)=\mu(A)=\eta(A \times X)$ for every Borel set $A$, but such that $\eta$ is not Lebesgue measure on $X \times X$.

(d) Let $\eta$ be as in part (c) and let $I, J \subset X$ be intervals of length $x$ and $y$ respectively. Show that

$$x+y-1 \leqslant \eta(I \times J) \leqslant \min \{x, y\}$$

(e) Let $X$ be as in part (c). Fix $d \geqslant 2$ and let $\Pi_{i}$ denote the projection $\Pi_{i}\left(x_{1}, \ldots, x_{d}\right)=\left(x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{d}\right)$ from $X^{d}$ to $X^{d-1}$. Construct a probability measure $\eta$ on $X^{d}$, such that the image under each $\Pi_{i}$ coincides with the $(d-1)$-dimensional Lebesgue measure, while $\eta$ itself is not the $d$-dimensional Lebesgue measure. $[$ Hint: Consider the following collection of $2 d-1$ independent random variables: $U_{1}, \ldots, U_{d}$ uniformly distributed on $\left[0, \frac{1}{2}\right]$, and $\varepsilon_{1}, \ldots, \varepsilon_{d-1}$ such that $\mathbb{P}\left(\varepsilon_{i}=1\right)=\mathbb{P}\left(\varepsilon_{i}=-1\right)=\frac{1}{2}$ for each $i .]$