Probability of Wiener process hitting a particular point at an independent stopping time Assume we have a stopping time $T$ that is independent of a Wiener process $W$. If $T$ were taking discrete values (let's say in $\mathbb{N}_0$), one can easily show (using the independence and the properties of $W$) that for any fixed $y \in \mathbb{R}$
\begin{equation*}
\mathbb{P}[W_T=y]=\sum_{k=0}^\infty \mathbb{P}[T=k]\underbrace{\mathbb{P}[W_k=y]}_{0}=0.
\end{equation*}
If $T$ is taking values on the whole positive real line, I'd do this:
$$\mathbb{P}[W_T=y]=\int_{0}^\infty \underbrace{\mathbb{P}[W_t=y]}_{0} dF_T(t)=0,$$
where $F_T$ is the distribution function of $T$.
I am just slightly worried about the transition from the countable to the uncountable case. I think that if I have an absolute continuous stopping time, I could approximate it with discrete times (independent of $W$) whose distributions converge to the distribution of $T$ and for which the first equation then holds and get the second equation as a limit.
Are there any problems with this that I did not realize?
 A: Let $(P_t)_{t\ge0}$ denote a regular conditional distribution of $W$ given $T$. This means that each $P_t$ is a probability measure on $C[0,\infty)$, the set of continuous functions from $[0,\infty)$ to $\mathbb{R}$ endowed with the $\sigma$-algebra induced by the coordinate projections, such that
$$
  P(W\in A,T \in B) = \int_B P_t(A) d Q(t)
$$
where $Q$ is the distribution of $T$. Your assumption that $T$ and $W$ are independent implies that $P_t$ has the same distribution as $W$ for all $t\ge0$. This is essentially the general version of the fact known from discrete distributions that independence implies that conditioning "doesn't do anything." We then obtain
$$
  P(W_T = y) = \int_0^\infty P(W_T = y | T = t) dQ(t) \\
  =\int_0^\infty P(W_t = y | T = t) dQ(t) \\
  =\int_0^\infty P_t(\{w|w_t = y\}) dQ(t) \\
  =\int_0^\infty P(W_t = y) dQ(t) = 0,
$$
as desired, where we in the last equality used the assumption that $P(T=0)=0$, and in the first equality used another result from the theory of regular conditional distributions, namely the substitution property, allowing us to substitute the value of $T$ into the variable being conditioned on.
