= Solution
The <quantile function> $X_\mu=F_\mu^{-1}$ is nondecreasing, hence a <Borel measurable function>. More explicitly, for every $x\in\mathbb R$,
$$
\{\omega\in(0,1):X_\mu(\omega)\leq x\}
=(0,F_\mu(x)]\cap(0,1).
$$
Thus $X_\mu$ is a <random variable>, and because Lebesgue measure is being used on $(0,1)$,
$$
\mathbb P(X_\mu\leq x)=F_\mu(x).
$$
Its <cumulative distribution function> is therefore $F_\mu$, so
$$
\boxed{\mathcal L(X_\mu)=\mu.}
$$
This is <inverse transform sampling>.
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