Consider $\{X_t\}$, a Cox process with $X_0=0$ and intensity $\lambda_t$.
Let $\tau$ be a stopping time on a compact interval $[0,T]$ such that $\tau\equiv\inf\{t>0:X_t=1\},$ (e.g. the first time a light bulb goes out).
Consider the comparison between $E[\mathbf{1}_{\{\tau\leq T\}}]$ and $E[\mathbf{1}_{\{\tau\leq T+1\}}]$, where the expectation is defined with respect to a well-defined probability space.
Is the following proof attempt valid? If not, where is it incorrect?
My claim: $E[\mathbf{1}_{\{\tau\leq T\}}]\leq E[\mathbf{1}_{\{\tau\leq T+1\}}]$.
Proof:
$E[\mathbf{1}_{\{\tau\leq T\}}]=\int \mathbf{1}_{\{\tau(\omega)\leq T\}}dP(\omega)=P(\{\tau\leq T\})$.
$E[\mathbf{1}_{\{\tau\leq T+1\}}]=\int \mathbf{1}_{\{\tau(\omega)\leq T+1\}}dP(\omega)=P(\{\tau\leq T+1\})$.
By monotonicity, $P(\{\tau\leq T\})\leq P(\{\tau\leq T+1\})$.
$\therefore E[\mathbf{1}_{\{\tau\leq T\}}]\leq E[\mathbf{1}_{\{\tau\leq T+1\}}]$.