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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?

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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 $$\{x: f_n(x) \not \to f(x)\} = \bigcup_{k \geq 1} \limsup (E_n^{k^{-1}})^c.$$

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