Why define $\bigcup_{n\in I} A_n=\emptyset,\bigcap_{n\in I} A_n=\Omega$ when $I$ is the empty set In the textbook I used on Probability and measure theory, a definition catches my eye:

If $I=\emptyset$, we define $\bigcup_{n\in I} A_n=\emptyset,\bigcap_{n\in I} A_n=\Omega$

I didn't draw attention to this definition initially, but this definition puzzles me a lot now. I don't get what it means to map an empty index set to a collection of a subset of $\Omega$. I also don't understand why the author would bother talking about the scenario where the index set is empty.
 A: Sorry if my argument here is somewhat circular but I rather avoid getting into the guts of logic and set theory and stay within what one calls "naive set theory".
Recall that membership to a set $A$ can be described in terms of logical statements. Thais is, if $P$ is a proposition that can be either true or false (not both simultaneously), then $A=\{x\in \Omega: P(x)\}$ denotes the collection of elements in $\Omega$ that that make P(x) a TRUE statement.
For example, consider the set of all writes of the 19th century and let $A$ denote the set of writers that wrote a novel in Russian.
Dostoyevsky satisfies the proposition wrote a novel in Russian, so Dostoyevsky is in $A$; Mark Twain (to my knowledge) did not ever write a novel in Russian, so Mark Twain is not in $A$.

No back to mathematics:
The intersection of a collection of sets $\{A_i:i\in I\}$ is defined as
$$\bigcap_{i\in I}A_i=\{x\in\Omega: \text{for every $i$, if}\, i\in I,\,\text{then}\quad x\in A_i\}$$
When $I=\emptyset$, any $x\in\Omega$ makes the statement "if $i\in I$, then $x\in A_i$" a true statement, for the stamens $(i\in I)$ is always FALSE (on account that $I$ has no elements) in this case. Recall the table for the logic operation for $p\Rightarrow q$, which is equivalent to $\sim(q\wedge(\sim p))$ (negation of $q$ and no $p$):

*

*"TRUE implies TRUE" is TRUE,


*"TRUE implies FALSE" is FALSE,


*"FALSE implies TRUE" is TRUE,


*and  "FALSE implies FLASE" is TRUE.
Similarly, the union of a collection of sets $\{A_i:i\in I\}$ is defined as
$$ \bigcup_{i\in I}A_i=\{x\in \Omega: \text{there exists $i$ such that}\,\,i\in I,\,\text{and}\,\, x\in A_i\}$$
When $I$ is empty, then for every element $x$ of $\Omega$, the statement “there exist $i\in I$ and $x\in A_i$“ is FALSE (on account that $I$ has no elements. So in this case,
$$ \bigcup_{i\in I}A_i=\emptyset$$
Here, recall the logic table for the logic operation $p \wedge q$. It is TRUE only when both statements $p$ and $q$ are TRUE.
