= Solution
Saying that $U$ has law $\mu$ means that $\mu$ is the <pushforward measure>
$$
\mu=\mathbb P\circ U^{-1},
\qquad
\mu(A)=\mathbb P(U\in A).
$$
For a real random variable, $\mu$ is a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$.
The pair $U,V$ consists of <independent random variables> when
$$
\mathbb P(U\in A,V\in B)
=\mathbb P(U\in A)\mathbb P(V\in B)
$$
for every pair of Borel sets $A,B$; equivalently, their joint law is the product of their marginal laws.
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