Uniform distribution on a sphere Consider the unit ball $S_n$ (centered at the origin) in $\mathbb{R}^n$ for $n \ge 1$ and a stochastic process $(X_t)_{t\ge 0}$ taking values in $\mathbb{R}^n$. 
Let $T = \inf\{t > 0 \colon X_t \in \partial S_n\}$ i.e. the first hitting time of the boundary of the unit circle. I know that the distribution of $X_T$ is uniform on $\partial S_n$ but I am unsure how to write this. Is it acceptable to write $$\mathbb{P}(X_T \in d\textbf{x}) = d\textbf{x}$$ for $\textbf{x} \in \partial S_n$?
 A: You have to understand that you are trying to write here is an equality between measures on the surface of the sphere. $\text{dx}$ usually denotes the standard or normalized Lebesgue measure. 
A better practice here would be to define first the uniform probability measure over the sphere, give it a name, for example $\text{d}\omega$. Some denote it $\sigma$, but IMO that raises the risk of it mistaking with the surface measure (i.e. $ \text{d}\omega\times 4\pi$ in $3d$).
Then you're using an abusive notation to denote by $\mathbb{P}(X_T \in \text{d}x)$ the other measure:
\begin{align}
\mathbb{P}(X_T \in \text{d}x) : \mathcal{B}(\partial S_n)  &\to \mathbb{R_+}\\
\mathcal{A} &\mapsto \mathbb{P}(X_T \in \mathcal{A})
\end{align}
where $\mathcal{B}(\partial S_n)$ is the set of the Borelian sets of $\partial S_n$.
This way for example you could write things like, although I wouldn't advise you to do it,:
\begin{equation}
\mathbb{P}(X_T \in \mathcal{A}) = \mathbb{P}(X_T \in \text{d}x)(\mathcal{A}) = \int_{\mathcal{A}} \mathbb{P}(X_T \in \text{d}x)\text{d}\omega = \int_{\mathcal{A}} \mathbb{P}(X_T \in \text{d}x)
\end{equation}
Then only the equality you wrote in your question actually makes sense, i.e. for your process:
\begin{equation}
\mathbb{P}(X_T \in \text{d}x) = \text{d}\omega
\end{equation}
--Btw I stick with your notation $S_n$ the unit ball but a more standard way would be to call it $B_n$, and its surface $S^{n-1}$.
