I have some vectors $\bf v_1,v_2,\dots,v_n$. The goal is to select a subset of these vectors based on some criteria. Let's consider each of those vectors are some kind of investment and my problem is to find the best portfolio e.g. a selection $\left\{3,10,25\right\}$ corresponding to $\bf \left\{v_3, v_{10}, v_{25}\right\}$. I like to develop a measure over such selections (aka. subsets) let say $\mu\Big(\left\{3,10,25\right\}\Big)$.
There is a correspondence between the index sets, $\left\{1,2,\dots,n\right\}$ and the pre-defined vectors. I'm new to measure theory and I want to consider this as my toy example. Is there a particular setup from here? The finite countable set $\left\{1,2,\dots,n\right\}$ and its $\sigma$-algebra as powerset over this set seems well defined. But I'm not sure if it would be possible to take care of the correspondence between only indexes and the vectors which I try to design measure $\mu$.
The measure $\mu: E \to [0, \infty]$ which $E$ is a particular subset of $\left\{1,2,\dots,n\right\}$ e.g. $\left\{3,10,25\right\}$.
One trivial measure is the counting measure for this but I define the measure $\mu$ by taking into account the content of vector space V which my vectors are comes from. The vectors are fixed but the whole space is meaningful. What I'm not sure if such correspondence (having using $\bf v_i$ when measure the selection contains $i$) be suitable in realm of measure theory? Is there a particular topic which take care of such scenarios?