Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that
$$\int_0^T\theta_u dS_u\ge -a$$
for a $a>0$. Furthermore $\mathcal{F}_t$ denotes a filtration satisfying the usual conditions. Now let $A\in\mathcal{F}_t$. Why do we have
$$\int_0^T\theta_u \mathbf1_AdS_u\ge -a\tag{1}$$ i.e. the integral is well defined and also bounded by $-a$ from below? That the integral is well-defined follows from:
$$E[\int_0^T\theta_u^2\mathbf1_Ad\langle S\rangle_u]\le E[\int_0^T\theta_u^2d\langle S\rangle_u]<\infty$$
But why is $(1)$ true? thanks in advance.