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The empirical measure is a probability measure

Statement

For every $n \geq 1$ , every measurable space $(Z, F)$ , and every observed tuple $z_{1}, \dots, z_{n} \in Z$ , the empirical measure $P_{n} = \frac{1}{n} \sum_{i = 1}^{n} δ_{z_{i}}$ from Empirical measure of observed units is a probability measure on $(Z, F)$ .

Facts & Assumptions

Given: A measurable space $(Z, F)$ , an integer $n \geq 1$ , and fixed observations $z_{1}, \dots, z_{n}$ .

[F1]

Empirical measure of observed units: $P_{n} (A) = \frac{1}{n} \sum_{i = 1}^{n} δ_{z_{i}} (A)$ for every $A \in F$ .

[F2]

A Dirac set function is a probability measure: each Dirac set function $δ_{z_{i}}$ is a probability measure on $(Z, F)$ ; in particular it is countably additive, is $0$ on $\emptyset$ , and assigns mass $1$ to $Z$ .

[F3]

Probability measures and probability spaces: a probability measure is a measure with total mass $1$ .

Proof

1.1F1F2algebra

For any $A \in F$ , the finite sum in [F1] is nonnegative by [F2]. Also $P_{n} (\emptyset) = 0$ and $P_{n} (Z) = \frac{1}{n} \sum_{i = 1}^{n} 1 = 1$ .

1.2F1F2algebra

Let $(A_{k})_{k \geq 1}$ be pairwise disjoint members of $F$ . For each fixed $i$ , at most one term $δ_{z_{i}} (A_{k})$ is nonzero, and countable additivity in [F2] gives $δ_{z_{i}} (⋃_{k} A_{k}) = \sum_{k} δ_{z_{i}} (A_{k})$ . Because the sum over $i$ is finite and there are at most $n$ nonzero terms in the resulting double sum, we may regroup to obtain $P_{n} (⋃_{k} A_{k}) = \frac{1}{n} \sum_{i = 1}^{n} \sum_{k} δ_{z_{i}} (A_{k}) = \sum_{k} \frac{1}{n} \sum_{i = 1}^{n} δ_{z_{i}} (A_{k}) = \sum_{k} P_{n} (A_{k})$ . Thus $P_{n}$ is countably additive.

2.1F3step 1.1step 1.2∎

By step 1.1, $P_{n}$ has total mass $1$ , and by step 1.2 it is countably additive and vanishes on the empty set. Therefore it is a measure of total mass $1$ , hence a probability measure by [F3].

Depends on

Used by

Size-n bootstrap from the empirical law Algorithm
Empirical-law bootstrap draws are conditionally IID Theorem

Cited to discharge well-definedness by Empirical measure of observed units.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §§1–2, pp. 1–5 (standard reference, not scraped)
B. Efron and R. Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Stanford LCS Technical Report 19 / BIO 101 (October 1985; full-text report copy) (standard reference, not scraped)
P.B. Stark, Statistics 240 Lecture Notes, Part 8: Resampling Methods, §1.1, PDF pp. 2–3 (standard reference, not scraped)