Let $T,U\geq 0$ be random variable with range with $T$ absolute continuous. Set $X=\min\{T,U\},\delta=I_{\{T\leq U\}}$ where $I_S$ is indicator function over set $S$. Define processes for $t\geq 0$, $N(t)=I_{X\leq t, \delta=1},N^U(t)=I_{X\leq t,\delta=0}$ and filtration of sigma algebra $\mathcal{F}_t=\sigma\{N(u),N^U(u),0\leq u\leq t\}$.
Assume equality $E[N(t+s)-N(t)\mid \mathcal{F}_t]=E[I_{\{t<X\leq t+s,\delta=0\}}\mid \mathcal{F}_t]$.
"Over $\mathcal{F}_t$ set $\{\forall u\in [0,t],N(u)=N^U(u)=0\}=\{X>t\}$, above conditional expectation must be constant over the set by conditional expectation $\mathcal{F}_t$ measurable."
It seems that the book is implying that $\{X>t+\epsilon\}$ is not in $\mathcal{F}_t$ for $\epsilon>0$.
$\textbf{Q}:$ How do I see constancy here? By definition of $N(t),N^U(t)$, the sigma algebra generated cannot generate $\{X\leq t+\epsilon\}$ for $\epsilon>0$?