Q. 2.5

Question

For any sequence of events $E 1, E 2, . . .,$ define a new sequence $F 1, F 2, . . .$ of disjoint events (that is, events such that $F i F j = Ø$ whenever $i \neq j$ ) such that for all $n \geq 1$ , $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$

Step-by-Step Solution

Verified

Answer

We have $\cup_{1}^{n} F_{i} = \cup_{1}^{n} E_{i}$ .

1Step 1 Given Information.

For any sequence of events $E 1, E 2, . . .,$ define a new sequence $F 1, F 2, . . .$ of disjoint events.

2Step 3 Explanation.

$F_{i} F_{j} = φ, i \neq j E_{n}; n \geq 1 E_{1} \subset E_{2} \subset . . . \subset E_{n} \subset E_{n + 1} \subset . . . \lim_{n \to \infty} E_{n} = \cupE_{i} E_{1} \supset E_{2} \supset . . . \supset E_{n} \supset E_{n + 1} \lim_{n \to \infty} E_{n} = \capE_{n}$

$F_{1}, F_{2}, . . . F_{1} F_{2} = φ, F_{1} \cap F_{2} = φ φ \subset E_{1} \subset E_{2} . . . \subset E_{n} φ \subset (E_{1} \cap φ^{c}) \subset (E_{2} \cap {E_{1}}^{c}) . . . \subset (E_{n} \cap E_{n - 1})$