User:A.M.R./sigma-algebra

Sigma-algebra.

Say, $X=\left\{2,3,5,7\right\}$. $A_{1}=\left\{2\right\}$ and $A_{2}=\left\{5\right\}$. Therefore, $\left\{\emptyset ,\left\{2\right\},\left\{5\right\},\left\{3,5,7\right\},\left\{2,3,7\right\},\left\{2,5\right\},\left\{3,7\right\},X\right\}$ is a σ-algebra over set X, that is closed under complementation and countable unions of its members & also contains empty set.

Measure

Now, A measure on X is a function which assigns a real number to subsets of X; this can be thought of as making precise a notion of 'size' or 'volume' for sets.

Random Variable

Random variable is different than the source of randomness or scenarios. Suppose, in case of dice rolling, $\left\{1,2,3,4,5,6\right\}$ is Ω or set of scenarios. w is each outcomes. Now, rv is what assigns values to different outcomes of an experiment. Here, rv X would be $X(w)=w$. But, in case of stock market, the abstract variable w can be the different situations of market which is constituted with different factors & may not be fully expressed mathematically. But, the random variable can assign these scenarios some values which can be stock prices.