proof of the début theorem

Let ${(ℱ)}_{t\in 𝕋}$ be a right-continuous filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on the measurable space $(\mathrm{\Omega },ℱ)$, It is assumed that $𝕋$ is a closed subset of $ℝ$ and that ${ℱ}_t$ is universally complete for each $t\in 𝕋$.

If $A\subseteq 𝕋\times \mathrm{\Omega }$ is a progressively measurable set, then we show that its début

$$D(A)=inf\{t\in 𝕋:(t,\omega )\in A\}$$

is a stopping time.

As $A$ is progressively measurable, the set $A\cap ((-\mathrm{\infty },t)\times \mathrm{\Omega })$ is $ℬ(𝕋)\times {ℱ}_t$-measurable. By the measurable projection theorem it follows that

is in ${ℱ}_t$. If there exists a sequence $t_n\in 𝕋$ with $t_n>t$ and $t_n\to t$, then

On the other hand, if $t$ is not a right limit point of $𝕋$ then

In either case, $\{D(A)\le t\}$ is in ${ℱ}_t$, so $D(A)$ is a stopping time.