Meaning behind Filter in Set Theory In a course in logic and set theory, we studied the concept of a Filter. We defined a filter $F \in P(S)$ on $S$ an equivalent of the following definition from Jech's Introduction to Set Theory:
(a) $S \in F$ and $\emptyset \notin F.$
(b) If $X\in F$ and $Y \in F$ then $X \cap Y \in F$.
(c) If $X \in F$ and $X \subseteq Y \subseteq S$, then $Y \in F$.
I am having trouble grasping this concept and it's meaning.
My question is, what is the intuition behind this definition, and why are these kinds of sets called filters?
Thanks

It's similar to the concept of "almost everywhere". Suppose to every subset $T\subseteq S,$ you write
$$
\mu(T) \begin{cases} =1 & \text{if } T\in F, \\ = 0 & \text{if } S\smallsetminus T\in F, \\ \text{is undefined} & \text{otherwise.} \end{cases}
$$
Then, according to the definition of "filter", you have
\begin{align}
& \mu(\varnothing) = 0 \\[6pt]
& \mu(S) = 1 \\[6pt]
& \text{If } \mu(T_1), \mu(T_2) \text{ both exist, and } T_1\cap T_2 = \varnothing, \\
& \text{then } \mu(T_1\cup T_2) = \mu(T_1) + \mu(T_2).
\end{align}
Saying $\{x\in S: P(x)\} \in F$ is the same as saying $P(x)$ for almost all $x\in S.$
