I have the following stopping time $$\tau_a = \inf\{t\ \vert\ Y_t = -a \text{ or } t>T\}$$ for $$Y_t = \int^t_0\mu(\omega,s)\,dB_s -\int_0^t\mu^2(\omega,s)\,ds$$
Why if $s\leqslant \tau_a$ and $\lambda\leqslant 0$ then $Y_s\lambda\leqslant a\vert\lambda\vert$?
And why does the continuity of $Y_t$ imply that $1_{[\tau_a<T]}\rightarrow0$?
($\mu$ is $L^2_\text{Loc}$)