Question

For any sequence of events $E1,E2,...,$define a new sequence$F1,F2,...$of disjoint events (that is, events such that $FiFj=Ø$whenever $i\ne j$) such that for all$n\ge 1$, $\underset{1}{\overset{n}{\cup }}{F}_i=\underset{1}{\overset{n}{\cup }}{E}_i$

Short answer

We have$\underset{1}{\overset{n}{\cup }}{F}_i=\underset{1}{\overset{n}{\cup }}{E}_i$.

Step-by-step solution

Given Information.

For any sequence of events $E1,E2,...,$define a new sequence localid="1649078122634" $F1,F2,...$of disjoint events.

Explanation.

$$\begin{gathered} F_i{F}_j=\phi ,i\ne j \\ {E}_n;n\ge 1 \\ {E}_1\subset E_2\subset ...\subset E_n\subset E_{n+1}\subset ... \\ \underset{n\to \infty }{\lim }{E}_n={\cup E}_i \\ {E}_1\supset E_2\supset ...\supset E_n\supset E_{n+1} \\ \underset{n\to \infty }{\lim }{E}_n={\cap E}_n \end{gathered}$$

$$\begin{gathered} F_1,F_2,...F_1{F}_2=\phi ,F_1\cap F_2=\phi \\ \phi \subset E_1\subset E_2...\subset E_n \\ \phi \subset (E_1\cap {\phi }^c)\subset (E_2\cap {E_1}^c)...\subset (E_n\cap E_{n-1}) \end{gathered}$$

$$\begin{gathered} F_1=E_1\cap {\phi }^c \\ {F}_2=E_2\cap {E^c}_1 \\ . \\ . \\ . \\ {F}_n=E_n\cap E_{n-1} \end{gathered}$$

$$\begin{gathered} F_1{F}_2=(E_1\cap {\phi }^c)\cap (E_2\cap {E_1}^c) \\ =\phi \cap {\phi }^c\cap E_2 \\ =E_1\cap E_1 \\ =\phi \end{gathered}$$

$$\begin{gathered} F_i{F}_j=(E_i\cap {E^c}_{i-1})\cap (E_j\cap {E^c}_{j-1}) \\ =E_i\cap {E^c}_{j-1} \\ =\phi \end{gathered}$$

$$\begin{gathered} {\cup F}_i=(E_1\cap {\phi }^c)\cup (E2\cap {E_1}^c)\cup ...\cup (E_n\cap {E_{n-1}}^c) \\ ={\cup E}_i \end{gathered}$$