This is about the use of notations in a paper (the main problem is different and I simplified it here, I don't need help on how to prove it):
Points $P_1, P_2,\ldots,P_N$ are placed in the circle $x^2+y^2=1$ on the $x-y$ plane according to a Uniform distribution. We measure the distance of each point to the origin, we flip a coin for each measurement, if we observe heads, we add $1/2$ error to the measurement, otherwise, we do not add an error. Let $d_i$ be the measured distance of the point $P_i$ to the origin.
Denote by $\mathbb{P}_u$ the probability that over all the realizations of locations of the points and the tosses of the coins, $\sum_{i=1}^{N}d_i<C$.
First question: To calculate $\mathbb{P}_u$, first I fix the locations of the points and find the probability that $\sum_{i=1}^{N}d_i<C$ holds. Obviously, this probability is a random variable that depends on the locations of the points. What is the proper notation for this random variable?
The notation has to indicate that this random variable depends on the locations of the nodes. Maybe I have to first define $(x_i,y_i)$ as the location of $P_i$ and then define $\mathcal{L}=\{(x_1,y_1),\ldots,(x_N,y_N)\}$ to be the set of the locations of the points, and then use $\mathbb{P}_u(\mathcal{L})$? Is there a better way?
Second question: The next step to calculate $\mathbb{P}_e$ is to take the expected value of $\mathbb{P_e}(\mathcal{L})$ with respect to the locations of the points. I used the $\mathbb{E}_{P}[\cdot]$ notation for the expected value over all the locations of the points: $$\mathbb{P}_e=\mathbb{E}_{P}[\mathbb{P_e}(\mathcal{L})]$$
This notation is not informative and I don't like it. It's like the condition is removed from the expectation. Can you introduce a better notation that includes the condition in the Expected value?
I would say that if you denote each of your points by $X_i$, and then define the true distance $D_i = \|X_i\|$, and error $E_i$ you can define the observed distance to be
$$ \widetilde D_i = D_i + E_i,$$
(feeling free to use any other letters of your choosing). Then your conditional probability would be
$$\mathbf P\left[ \textstyle\sum_{i=1}^N \widetilde D_i < C \, \big| \, X_1=\underline x_1,\ldots, X_N=\underline x_N\right].$$
The probability (not conditioned on the locations of the points) would then be
$$\mathbf P\left[ \textstyle \sum_{i=1}^N \widetilde D_i < C \right] = \int \mathbf P\left[ \textstyle\sum_{i=1}^N \widetilde D_i < C \, \big| \, X_1=\underline x_1,\ldots, X_N=\underline x_N\right] f_{X_1,\ldots, X_N}(\underline x_1,\ldots, \underline x_N) d\underline x_1 \cdots d \underline x_N $$ where $f_{X_1,\ldots,X_N}$ is the joint density function of the points; if they are i.i.d. this becomes $$f_{X_1,\ldots,X_N}(\underline x_1, \ldots, \underline x_N) = \prod_{i=1}^N f_X(\underline x_i),$$ with $f_X$ the pdf of a single point.
In response to the comments below:
Beyond pointing out what the standard notation would be, I do not know how else I can help you.
If you want something succinct you could define certain events; i.e. define $$A_{N,C} = \left\{\textstyle \sum_{i=1}^N \widetilde D_i < C\right\},$$ and then reduce $\{X_1 = \underline x_1, \ldots, X_N = \underline x_N\}$ to $ \{\underline X = \underline x\}$, where now the interpretation of the underline has changed. i.e. $\underline X = (X_1,\ldots, X_N)$ and $\underline x = (x_1,\ldots, x_N)$, so in particular $x_i \in \mathbf{R}^2$ is now a vector, but doesn't have an underline. So now your probability would be
$$ \mathbf{P} \left[ A_{N,C} \, | \, \underline X = \underline x\right].$$
I think avoiding the conditional probability notation (i.e. $\mathbf P[ \cdot | \cdot ]$) is a bad idea as this is standard notation in probability, and will help to make your work consistent / clear to your audience.