A type of set used in convergence in measure theory

This is not a specific problem, but a general question. Often when we're showing convergence of functions (particularly pointwise) or even of sets in certain cases, a set of the following form appears: $$\bigcap_{N \in \mathbb{N}} \bigcup_{n \geq N} E_{n}$$ Often we will encounter situations where we want to show that something goes "wrong" on a set of this form, or a set of this form is measurable (which is obviously true if $$E_{n}$$ are measurable); I have a general question about a set of this form.

In what contexts does this show up? In my understanding, something is in this set if it "happens" infinitely often. This is a slightly probabilistic interpretation, but if some property holds for infinitely many sets, then something like this shows up. Is that a correct interpretation? I'd appreciate some elaboration.

Example: If we have a sequence of functions $$\{f_{n}\}$$, where each $$f_{n}$$ is supported on a measurable set $$E_{n}$$, then

$$\bigcap_{N} \bigcup_{n \geq N} E_{n}$$

is precisely the set of points where the sequence $$f_{n}$$ is not eventually $$0$$. Thus, points $$x$$ at which arbitrarily many $$f_{n}$$ are non-zero (points in infinitely many supports $$E_{n}$$). Is this correct?

What you've written is called $$\limsup E_n$$ and $$x \in \limsup E_n$$ iff $$x \in E_n$$ for infinitely many $$n$$. This understanding already shows your example is correct.
There is a similar notion that $$\liminf E_n = \bigcup_{j \geq 1} \bigcap_{n \geq j} E_n$$. $$x \in \liminf E_n$$ iff there is an $$N$$ such that $$x \in E_n$$ whenever $$n \geq N$$ (i.e. "$$x \in E_n$$ eventually").
These sets occur naturally when considering convergence as you noted. In fact, $$\liminf E_n$$ is defined to capture definitions like that of convergence which require some behaviour to occur "eventually". Then $$\limsup E_n$$ is just the complement of $$\liminf E_n$$.
For example, in the case of convergence we require that for fixed $$\varepsilon > 0$$, your sequence lies within distance $$\varepsilon$$ of the limit "eventually". More precisely, if $$E_n^\varepsilon = \{x: |f_n(x) - f(x)| < \varepsilon\}$$ then $$\liminf E_n^\varepsilon = \{x: \exists N \text{ such that } \forall n \geq N, |f_n(x) - f(x)| < \varepsilon\}$$ so that we can write $$\{x: f_n(x) \to f(x)\} = \bigcap_{k \geq 1} \liminf E_n^{k^{-1}}.$$
Conversely, if we want to find $$x$$ such that $$f_n(x) \not \to f(x)$$ then there is an $$\varepsilon > 0$$ such that $$|f_n(x) - f(x)| > \varepsilon$$ infinitely often which is the same as saying $$x \in \limsup (E_n^\varepsilon)^c$$. Hence we can write $$\{x: f_n(x) \not \to f(x)\} = \bigcup_{k \geq 1} \limsup (E_n^{k^{-1}})^c.$$