Question about subsets of Independent random variables and events

Definition for independent random variables:

The countably valued random variables $$X1,...,X_n$$ are said to be independent iff for numbers $$x_1,...x_n,$$ we have $$P(X_1=x_1,...,X_n=x_n)=P(X_1=x_1)...P(X_n=x_n)$$

Deduce further by replacing single values $$x_i$$ by arbitrary sets $$S_i$$:

For arbitrary countable sets $$S_1,...S_n:$$ $$P(X_1\in S_1,...,X_n\in S_n)=P(X_1\in S_1)...P(X_n\in S_n)$$

Definition for independent events:

the n events $$A_1,A_2,...,A_n$$ are independent if the intersection of any subset of them has its probabilities of the individual events.

For any subset$$(i_1,i_2,...,i_k) \text{ of } (1,2,...,n):$$ $$P(A_{i_1}∩A_{i_2}∩...∩A_{i_k})=P(A_{i_1})P(A_{i_2})...P(A_{i_n})$$

my question: For indenpent random variables, does the property still apply on any of the subsets? if not provide an example.

• Yes it still holds Aug 5, 2020 at 19:14

Yes.  Just let $$A_i$$ be the event that $$X_i\in S_i$$. Any countable set of such events— $${\{X_i\in S_i\}}_{i=1}^n$$ for any collection of arbitrary set of values $${\{S_i\}}_{i=1}^n$$ —shall be mutually independent exactly when the corresponding indexed set of random variables– $${\{X_i\}}_{i=1}^n$$ –are mutually independent.