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.