---
tag: 4SSJ589G6T
kind: statement
role: theorem
name: Portmanteau theorem
msc: [60B10, 28A33]
uses: [MBW9JCGZC1, 0VKD1TP7F3, 059TJN0NZ1, 0ZCK8DQYN1, 02BH3MCBQB, 0AW1E6YTJY, 0T3RZQXDYD]
about: [MBW9JCGZC1]
sources:
  - url: https://en.wikipedia.org/wiki/Convergence_of_measures
---

## Statement

Let $X$ be a [[metric space|0VKD1TP7F3]] and let $\mu, \mu_1, \mu_2, \dots$ be Borel [[probability measures|059TJN0NZ1]] on $X$.
For a set $A \subseteq X$, let $\partial A$ be its boundary.
The following conditions are equivalent.

1. $\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\mu$: $\int f \, d\mu_n \to \int f \, d\mu$ for every bounded continuous $f \colon X \to \R$.
2. $\int f \, d\mu_n \to \int f \, d\mu$ for every bounded [[Lipschitz|0AW1E6YTJY]] function $f \colon X \to \R$.
3. $\limsup_{n \to \infty} \mu_n(F) \le \mu(F)$ for every [[closed set|0ZCK8DQYN1]] $F \subseteq X$.
4. $\liminf_{n \to \infty} \mu_n(G) \ge \mu(G)$ for every [[open set|02BH3MCBQB]] $G \subseteq X$.
5. $\mu_n(A) \to \mu(A)$ for every Borel set $A$ with $\mu(\partial A) = 0$.

## Remarks

Condition 1 implies 2 at once.
For 2 implies 3, the Lipschitz functions $f_k(x) = \max(0, 1 - k \, d(x, F))$ decrease to the indicator of $F$ as $k \to \infty$.
Conditions 3 and 4 are equivalent by taking complements.
Conditions 3 and 4 applied to the closure and the interior of $A$ give 5.
For 5 implies 1, write $\int f \, d\mu$ as an integral of $\mu(\set{f > t})$ over $t$; for all but countably many $t$ the set $\set{f > t}$ has a $\mu$-null boundary.

Example: $\delta_{1/n}$ converges weakly to $\delta_0$ on $\R$.
For the open set $G = (0, 1)$, one has $\delta_{1/n}(G) = 1$ for $n \ge 2$ and $\delta_0(G) = 0$, so the inequality in 4 can be strict.
For $A = \set{0}$, the boundary is $\set{0}$, which has $\delta_0$-measure $1$.
So condition 5 makes no claim, and indeed $\delta_{1/n}(A) = 0$ does not tend to $\delta_0(A) = 1$.

## Explanations

### L1 human

The theorem lists several equivalent ways to say that measures converge weakly.
In the limit, a closed set can gain mass and an open set can lose mass, but not the other way round.
Sets whose boundary carries no limit mass have converging measures.

### L2 human

One uses whichever form is easiest to check.
Condition 2 is handy because Lipschitz functions are easy to control, while condition 5 gives convergence of probabilities of concrete events.
For real random variables, condition 5 applied to half-lines gives convergence of distribution functions at continuity points.
The theorem is a standard tool together with Prokhorov's theorem and the continuous mapping theorem.