I'm new to formal writing, so please be patient :).
Let $S=\left\{ s \subset [0,1] : |s|=n \right\}$, and let $x$ be a probability distribution over the elements of $S$. Namely, $x$ is a distribution over all sets of size $n$ (for a constant $n$) of elements from $[0,1]$.
Define $\mu: 2^{[0,1]}\rightarrow [0,n]$ such that for every Lebesgue measurable set $A\subset [0,1]$ $$\mu(A) = \int_{S}|s\cap A|dx(s)=\mathbb E_{s\sim x}(|s\cap A|) $$
(1) How can one show that $([0,1],\mathcal B([0,1]),\mu)$ is a measure space?
(2) Am I using the right notations? is there a better introduction of this problem?
Referring to (1), I thought about the following:
- sort every element $s$ to obtain a tuple,$(s_1,\dots,s_n)\in [0,1]^n$.
- look at the projection of $s$ on each of the components. If the projection on every component $i$, $P_i$, is itself a measure, $\mu$ can be expressed as sum of indicators, $$\mu(A) =\sum_{i=1}^n\int_{S}\mathbb 1_{s_i \in A}dP_i(s_i), $$ and since countable sum of measures is itself a measure (see, e.g., here) we are done.
I'm not sure though that this is right. Another option is to state straightforward that $$\mu(A) =\int_{S}\sum_{i=1}^n\mathbb 1_{s_i \in A}dx(s), $$ and to flip the order of summation, which is possible due to non-negativity using Fubini's theorem.
Any ideas?
