Let $\Omega$ be an uncountable set and $(\Omega, \mathcal{F},P)$ be a probability space built on $\Omega$. 


Let $S \subset \{A \in \mathcal{F}: P(A)=0,\;|A|=1\}:|S|<\infty$ be a *finite* subset of the class of singleton P-null sets in $\mathcal{F}$.


I am trying to use $S$ to construct a new probability space from $(\Omega, \mathcal{F},P)$: $(S,\sigma(S),P')$


Since $\sigma(S)$ is discrete, I can define a potential probability measure for $P'$ on the above space by specifying its values for each sample point in $S$: $P'(s\_i)=\lim\limits\_{n\rightarrow \infty} \frac{P(Q\_{in})}{ P \left(\bigcup\limits\_{j \in \{1...|S|\}}Q\_{jn}\right) }$, where $Q\_{in} \subset \mathcal{F}:Q\_{ij}\supset Q\_{ik}\;\forall( j\leq k)$ and $\lim \inf Q\_{in} = s\_i\in S$ 


Note that $P'$ can be extended to $\sigma(S)$ via additivity.


**Question 1:** Under what conditions does $P'$ exist?


**Question 2:** What additional conditions are needed for the above to be true if $|S| = \infty$? 




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**My thinking so far**


The use of the limit in $P'$ requires a metric space $(\Omega,d)$, such that $P(B\_r(s) :=\{\omega \in \Omega: d(s,\omega)<r \})>0 \;\forall(s\in S, r>0)$, which allows the above limit to define a probability measure on $\sigma(S)$. However, I'm not sure if this condition is sufficient, necessary or both?


I initially developed the above notions by working with the simple probability space $([0,1],\mathcal{B}([0,1]),P(A\subset \mathcal{B}([0,1]) = \lambda(A))$. If $|S|=M$, then we have a set of $M$ distinct points in $[0,1]$. We can define $Q\_{in} := [(s\_i - \frac{s\_i}{n}),(s\_i+\frac{1-s\_i}{n})] \;\forall i:s\_i \in S$. This results in a possible candidate for $P'$:


$P'(s\_i) = \lim \limits\_{n\rightarrow \infty} \frac{\lambda([(s\_i - \frac{s\_i}{n}),(s\_i+\frac{1-s\_i}{n})])}{ \lambda \left(\bigcup\limits\_{j \in \{1...|S|\}}[(s\_i - \frac{s\_j}{n}),(s\_j+\frac{1-s\_j}{n})]\right) }$.


The numerator is easy to calculate for all $n$; however, the $Q\_{in}$ are not initially disjoint so the denominator is complicated at the beginning of the sequence. However, since the elements of $S$ are countable and distinct, $\exists n\_0: Q\_{in}\cap Q\_{jn} = \emptyset \; \forall (i\neq j, n>n\_0)$. Therefore, we can restrict analysis of the above limit to $n>n\_0$ without loss of generality. The benefit of doing so is that the union of the $Q\_{in}$ becomes a *disjoint* union, and we can get a simple formula for the denominator. Specifically,


$P'(s\_i) = \lim \limits\_{n\rightarrow \infty} \frac{\lambda([(s\_i - \frac{s\_i}{n\_0+n}),(s\_i+\frac{1-s\_i}{n\_0+n})])}{ \lambda \left(\biguplus\limits\_{j \in \{1...|S|\}} [(s\_i - \frac{s\_j}{n\_0+n}),(s\_j+\frac{1-s\_j}{n\_0+n})]\right) } = \frac{1/(n\_0+n)}{M/(n\_0+n)} = \frac{1}{M}\; \forall s\_i \in S$


Thus, in this very simple example, $P'$ converges at $n\_0<\infty$.


It seems straightforward to extend this to a countable set on the domain of a non-uniform distribution function F (e.g., gaussian). The denominator is guaranteed to be $\leq 1$ since F is a distribution; therefore, the denominator will always be a subset of the domain of F and the numerator, being a subset of the union in the denominator, will always be $\leq$ the denominator. Therefore, it seems like this is quite general, since all random variables map to the real numbers (hence we have a metric space).


I am not sure if I am missing some, possibly pathological case, where you cannot define $P'$ from $P$ 



Using a probability measure, P, defined on uncountable sets to construct a probability measure, P' on singleton P-null sets
The condition you have is not sufficient. Although the question has some wording issues, I can answer it as I think you intended it.


Suppose $(\mathbb{R},\mathcal{B},P)$ is our probability triple, where $P([0,\frac{1}{2^n}])=4^{-\lceil\frac{n}{2}\rceil}$. In this case, there can be no limit defining $P'(\{0\})$ if the sets you are using in your limit are balls size 2^{-n}$.


A simpler counterexample (that you may be more or less happy with):


$P$ has probability density function $f(x)=\left\{\begin{array}\,e^{-x}&x>0\\0&x<0\end{array}\right.$. In this case, depending on whether the sets tending down to zero are to the left or the right, you will get different answers for $P'(\{0\})$


Note: You can always define some $P'$ based on some specific sequence $(Q\_{in})\_{i=1,n=1}^{i=|S|,n=\infty}$ using your definition; it just isn't unique or useful. The way to define it in general is to first choose any sequence $Q\_{in}$ with $P(Q\_{in})>0$ for all $i$ and $n$, and use Bolzano-Weierstrauss to find a convergent subsequence of the sequence of $|S|$-tuplets $P(Q\_{in})$, and name this sequence (Q^\*\_{in}). Then define $Q^{\*,1}\_{in}=Q^\*\_{in}$ and define P'\_1:{{s\_1}}\rightarrow [0,1] by $P'\_1({s\_1})=1$.


Inductively define $P'\_k:\{\{s\_i\}|i\le k\}\rightarrow [0,1]$ by choosing an $s\_i$ with $P'\_{k-1}(s\_i)\ne 0$ and a subsequence $Q^{\*,k}\_{in}$ of $Q^{\*,k-1}\_{in}$ s.t. $\frac{P(Q^{\*,k}\_{in})}{P(Q^{\*,k}\_{kn})}$ has a limit in $[0,\infty]$. Then we can define $P'\_k(s\_i)$ just by keeping everything in the appropriate ratios. $P'\_{|S|}$ will be a probability distribution defined by your limit from the sequence of sets $(Q^{\*,|S|}\_{in})\_{i=1,n=1}^{i=|S|,n=\infty}$


