= Solution
On a product extension of the <probability space>, take a <uniform distribution> variable $U$ on $(0,1)$ independent of the original <sigma-algebra>. For $x>0$,
$$
\mathbb P(M_0/U\geq x\mid\mathcal F_0)
=\mathbb P(U\leq M_0/x\mid\mathcal F_0)
=1\wedge\frac{M_0}{x}.
$$
These are the same conditional tails as in the <maximal identity for a continuous nonnegative local martingale tending to zero>. Taking <expectations> identifies the unconditional <probability distributions>:
$$
\boxed{M^*\ \stackrel{d}{=}\ M_0/U.}
$$
If $M_0=0$, define $M_0/U=0$; the endpoint $U=0$ is a null event if one uses $[0,1]$ instead. The possible atom at zero has mass $\mathbb P(M_0=0)$, since the conditional probability of a positive maximum is zero exactly on that event. For fixed $M_0=m>0$, the conditional law is the <Pareto distribution> of shape one and minimum $m$, with density $m/x^2$ on $x>m$.
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