You've seen the notation. Maybe in a probability textbook. Maybe in a measure theory lecture where the professor wrote $\limsup A_n$ and $\liminf A_n$ on the board like everyone should just know what they mean Not complicated — just consistent..
And if you're like most people, you nodded along. Then you went home, stared at the definitions, and thought: wait, what does "infinitely often" actually mean here?
Here's the thing — these concepts aren't mysterious. They're just set theory's way of talking about patterns that persist. And once you see them visually, they click.
What Are limsup and liminf of Sets
Let's start with a sequence of sets. Consider this: $A_1, A_2, A_3, \dots$ — each one a subset of some universal space $\Omega$. But could be events in a probability space. Could be measurable sets in $\mathbb{R}$. The setting doesn't change the definitions Simple, but easy to overlook. Simple as that..
The Formal Definitions
$\liminf_{n \to \infty} A_n = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k$
$\limsup_{n \to \infty} A_n = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k$
Read them slowly. But the liminf is a union of intersections. The limsup is an intersection of unions. That order matters — a lot.
What They Actually Mean
liminf $A_n$ — the set of points that belong to all but finitely many $A_n$. Eventually, they're in every set. They stick around The details matter here. That's the whole idea..
limsup $A_n$ — the set of points that belong to infinitely many $A_n$. They keep showing up. Maybe they leave. Maybe they come back. But they return infinitely often Worth keeping that in mind..
That's it. Think about it: that's the whole intuition. Everything else follows from those two sentences.
A Visual Way to Think About It
Picture a point $\omega \in \Omega$. Track its membership across the sequence:
- $\omega \in A_1$? Yes
- $\omega \in A_2$? No
- $\omega \in A_3$? Yes
- $\omega \in A_4$? Yes
- $\omega \in A_5$? No
- ...
If the "Yes" answers eventually become permanent — after some index $N$, it's always yes — then $\omega \in \liminf A_n$.
If the "Yes" answers keep appearing forever, even with gaps — then $\omega \in \limsup A_n$.
Every point in liminf is automatically in limsup. The reverse isn't true.
$\liminf A_n \subseteq \limsup A_n$
Always. No exceptions.
Why This Matters
You might wonder: why do we care about "eventually always" versus "infinitely often"?
Because probability theory runs on this distinction Not complicated — just consistent..
The Borel-Cantelli Lemmas
This is where limsup of sets earns its keep. The first Borel-Cantelli lemma says: if $\sum P(A_n) < \infty$, then $P(\limsup A_n) = 0$.
Translation: if the total probability mass of your events is finite, then almost surely only finitely many of them occur. The "infinitely often" set has probability zero.
The second lemma (with independence): if $\sum P(A_n) = \infty$ and the events are independent, then $P(\limsup A_n) = 1$.
Infinitely many occur almost surely That's the whole idea..
This isn't abstract. It's the foundation for:
- Almost sure convergence of random variables
- The law of the iterated logarithm
- Zero-one laws
- Ergodic theory
Measure Theory and Integration
In measure theory, liminf and limsup of sets give you Fatou's lemma for sets:
$\mu(\liminf A_n) \leq \liminf \mu(A_n)$ $\mu(\limsup A_n) \geq \limsup \mu(A_n)$
These inequalities are the set-theoretic ancestors of the integral versions. They tell you that measure behaves "continuously" with respect to these limits — but only in one direction each way Worth knowing..
When the Limit Exists
If $\liminf A_n = \limsup A_n$, we say the sequence converges and write $\lim A_n$ for the common value.
This happens exactly when every point is either eventually always in the sets or eventually always out. No oscillation allowed And that's really what it comes down to..
How to Work With Them
Computing liminf and limsup: A Step-by-Step Approach
Let's do this concretely. Suppose $\Omega = \mathbb{R}$ and:
$A_n = \begin{cases} [0, 1 + \frac{1}{n}] & \text{if } n \text{ is odd} \ [0, 1 - \frac{1}{n}] & \text{if } n \text{ is even} \end{cases}$
Step 1: Understand the pattern. Odd-indexed sets are slightly larger than $[0,1]$. Even-indexed sets are slightly smaller.
Step 2: Find $\bigcap_{k=n}^{\infty} A_k$ for a fixed $n$. This is the set of points in every $A_k$ from $n$ onward It's one of those things that adds up..
For large $n$, the even sets shrink toward $[0,1)$ and odd sets shrink toward $[0,1]$. The intersection will be $[0, 1 - \frac{1}{n}]$ if $n$ is even, or $[0, 1 - \frac{1}{n+1}]$ if $n$ is odd. Either way, as $n$ grows, these intersections approach $[0,1)$ Most people skip this — try not to..
Step 3: Take the union over $n$. $\bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} A_k = [0, 1)$ Most people skip this — try not to. No workaround needed..
So $\liminf A_n = [0, 1)$.
Step 4: Find $\bigcup_{k=n}^{\infty} A_k$ for a fixed $n$. This is points in at least one $A_k$ from $n$ onward And that's really what it comes down to..
For any $n$, there are odd indices ahead, so the union always includes $[0, 1 + \frac{1}{n}]$ (or slightly larger). As $n$ grows, these unions shrink toward $[0,1]$ The details matter here..
Step 5: Take the intersection over $n$. $\bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k = [0, 1]$.
So $\limsup A_n = [0, 1]$.
Notice: $\liminf \neq \limsup$. The sequence doesn't converge. Points in $(1, 1+\epsilon)$ appear infinitely often (in odd sets) but not eventually always.
The Indicator Function Trick
Here's a technique that saves enormous time. For any $\omega \in \Omega$, define the indicator sequence:
$x_n(\omega) = \mathbf{1}_{A_n}(\omega) = \begin{cases} 1 & \omega \in A_n \ 0 & \omega \notin A_n \end{cases}$
Then:
$\mathbf{1}{\liminf A_n}(\omega) = \liminf{n \to \infty} x_n(\omega)$ $\mathbf{1}{\limsup A_n}(\omega) = \limsup{n \to \infty} x_n(\omega)$
The liminf/limsup of sets is the pointwise liminf/limsup of the indicator functions Which is the point..
This means you can compute set limits
Leveraging Indicator Functions for Practical Computation
The indicator‑function trick turns the problem of finding set‑wise limits into a pointwise problem that is often far easier to handle. Consider this: in essence, the liminf and limsup of a sequence of measurable sets are exactly the sets of points whose indicator sequences have the corresponding pointwise liminf (resp. limsup) equal to 1.
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Identify the pointwise behaviour.
For each (\omega\in\Omega) consider the binary sequence (x_n(\omega)=\mathbf 1_{A_n}(\omega)). Determine (\liminf_{n}x_n(\omega)) and (\limsup_{n}x_n(\omega)). These are simply the smallest and largest accumulation points of the 0–1 sequence That's the whole idea.. -
Translate back to sets.
(\displaystyle \liminf_{n\to\infty}A_n={\omega:\liminf_{n}x_n(\omega)=1}) and
(\displaystyle \limsup_{n\to\infty}A_n={\omega:\limsup_{n}x_n(\omega)=1}).
Because the values are only 0 or 1, the pointwise limits are easy to read off: a point belongs to the liminf exactly when the indicator is eventually constantly 1; it belongs to the limsup exactly when the indicator is infinitely often 1 Still holds up..
Example: A “wiggling” Interval Sequence
Let (\Omega=[0,2]) and define
[ B_n= \begin{cases} \bigl[0,1+\tfrac{1}{n}\bigr) & n\ \text{odd},\[4pt] \bigl( \tfrac12,1-\tfrac{1}{n}\bigr] & n\ \text{even}. \end{cases} ]
The odd sets stretch beyond 1, while the even sets contract to a sub‑interval that stays strictly below 1. Using the indicator trick:
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For (\omega\le \tfrac12) we have (\mathbf 1_{B_n}(\omega)=1) for all (n) (the even sets still contain (\tfrac12)). Hence (\liminf=\limsup) contains ([0,\tfrac12]) The details matter here..
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For (\omega\in(\tfrac12,1)) the indicator oscillates: odd (n) give 1, even (n) give 1 only when (\omega>1-\tfrac{1}{n}). As (n\to\infty) the even‑index condition eventually fails, so the point is 1 infinitely often but not eventually always. Thus (\omega\in\limsup) but (\omega\notin\liminf).
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For (\omega=1) the odd sets contain it, the even sets exclude it for all sufficiently large even (n); again (\omega\in\limsup\setminus\liminf).
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For (\omega>1) the odd sets contain it only when (\omega<1+\tfrac{1}{n}); this holds for infinitely many odd (n) but never eventually. Hence (\omega\in\limsup) as well.
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Points (\omega\ge 2) never appear, so they are outside both limits Small thing, real impact..
Putting these observations together,
[ \liminf_{n\to\infty}B_n=[0,\tfrac12],\qquad \limsup_{n\to\infty}B_n=[0,1)\cup{!1!}=[0,1]. ]
Notice how the indicator viewpoint makes the alternating inclusion/exclusion transparent without having to manipulate nested intersections and unions directly Still holds up..
Why the Trick Works: A Short Proof
Let (x_n(\omega)=\mathbf 1_{A_n}(\omega)). By definition of set liminf,
[ \omega\in\liminf_{n}A_n \iff \exists N;\forall n\ge N:;\omega\in A_n \iff \exists N;\forall n\ge N:;x_n(\omega)=1 \iff \liminf_{n}x_n(\omega)=1 . ]
The second equivalence uses the elementary fact that for a 0–1 sequence, “eventually constantly 1’’ is exactly the definition of the liminf being 1. An analogous argument yields the limsup statement. Hence the indicator representation is not merely a convenience—it is an exact equality of sets.
Extensions and Applications
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Convergence of measurable sets. When (\liminf A_n=\limsup A_n) we say the sequence converges (in the sense of sets). The indicator trick shows that convergence is equivalent to pointwise convergence of the corresponding indicator functions, which is precisely the notion of convergence used in measure theory and probability.
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Egorov’s theorem and convergence almost everywhere. Because
Because the indicator functions ( \mathbf 1_{A_n} ) take only the values 0 and 1, pointwise convergence of these functions is equivalent to the set‑theoretic statement that each point eventually belongs to (or eventually stays out of) the sets (A_n). Also, when this convergence holds almost everywhere, Egorov’s theorem guarantees that, for any (\varepsilon>0), there exists a measurable set (E) with (\mu(E)<\varepsilon) on which the convergence is actually uniform. Translating back to sets, uniform convergence of the indicators means that there is an index (N) such that for all (n\ge N) the symmetric difference (A_n\triangle A) has measure less than (\varepsilon); in other words, the sequence ((A_n)) converges to (A) in measure. Thus the indicator trick not only clarifies liminf and limsup but also bridges the three fundamental modes of convergence for sequences of measurable sets: pointwise (set) convergence, almost‑everywhere convergence, and convergence in measure Worth keeping that in mind..
A second important application appears in the Borel–Cantelli lemmas. Using indicators, (\limsup A_n={\omega:\limsup_n \mathbf 1_{A_n}(\omega)=1}); the bound on the sum of measures yields (\int \limsup_n \mathbf 1_{A_n},d\mu\le \sum_n\int \mathbf 1_{A_n},d\mu<\infty), forcing the limsup to be zero almost everywhere. Consider this: the first lemma states that if (\sum_n \mu(A_n)<\infty) then (\mu(\limsup A_n)=0). The second lemma (the converse under independence) follows similarly by examining the divergence of the series and applying the second Borel–Cantelli lemma to the indicator variables That's the part that actually makes a difference..
Finally, Fatou’s lemma for non‑negative measurable functions is an immediate corollary: for any sequence (f_n\ge 0),
[ \int \liminf_n f_n,d\mu\le \liminf_n \int f_n,d\mu, ]
which is obtained by applying the set‑level inequality to the sets ({f_n>t}) and integrating over (t\ge 0) (the “layer‑cake’’ representation). Here again the indicator trick reduces a functional statement to a purely set‑theoretic one.
To keep it short, replacing a sequence of sets by its sequence of indicator functions converts abstract liminf and limsup operations into familiar pointwise limits of 0‑1 sequences. This simple reinterpretation makes proofs transparent, connects set convergence with function convergence, and underlies cornerstone results such as Egorov’s theorem, the Borel–Cantelli lemmas, and Fatou’s lemma. The indicator viewpoint is therefore not merely a notational convenience but a powerful unifying tool in measure theory and probability Nothing fancy..
