6. For any closed interval $I$, let $M_{I}$ denote the maximum value of the function $y=\sin x$ on $I$. If the positive number $a$ satisfies $M_{[0, a]}=2 M_{[a, 2 a]}$, then the value of $a$ is $\qquad$ .
6. $\frac{5 \pi}{6}$ or $\frac{13 \pi}{12}$.

If $0 < a < \frac{\pi}{2}$, then $M_{[0, a]}=1$.
Thus, $M_{[a, 2 a]}=\frac{1}{2}$.
Therefore, there exists a non-negative integer $k$ such that
$$
2 k \pi + \frac{5 \pi}{6} \leqslant a < 2 a \leqslant 2 k \pi + \frac{13 \pi}{6},
$$

and at least one of the two “*” in equation (1) must be an equality.
When $k=0$, we get $a=\frac{5 \pi}{6}$ or $2 a=\frac{13 \pi}{6}$.
Upon verification, $a=\frac{5 \pi}{6}$ and $\frac{13 \pi}{12}$ both satisfy the conditions.
When $k \geqslant 1$, by
$$
2 k \pi + \frac{13 \pi}{6} < 2\left(2 k \pi + \frac{5 \pi}{6}\right),
$$

we know there is no $a$ that satisfies equation (1).
In summary, the value of $a$ is $\frac{5 \pi}{6}$ or $\frac{13 \pi}{12}$.