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A finite empirical law is a probability measure

Statement

For any $n \geq 1$ and $x_{1}, \dots, x_{n} \in R^{d}$ , the set function $P_{n} (A) = \frac{1}{n} \sum_{i = 1}^{n} 1 {x_{i} \in A}, A \in B (R^{d}),$ is a probability measure.

Facts & Assumptions

Given: The finite data list and empirical set function in the statement.

[F1]

The empirical set function is the finite average of the point indicators. IID empirical bootstrap sample and mean statistic

[F2]

A measure assigns zero to the empty set and is countably additive on pairwise disjoint measurable sets. Measures on sigma-algebras

[F3]

A probability measure is a measure whose total mass is one. Probability measures and probability spaces

[F4]

A nonnegative extended series is the supremum of its finite partial sums. Series in the nonnegative extended real line

Proof

1.1F1given

Formula [F1] gives $P_{n} (\emptyset) = 0$ and $P_{n} (R^{d}) = n^{- 1} \sum_{i = 1}^{n} 1 = 1$ .

1.2F1F4givenalgebra

Let $(A_{k})_{k \geq 1}$ be pairwise disjoint Borel sets. For each fixed $i$ , at most one indicator $1 {x_{i} \in A_{k}}$ is nonzero. Therefore $P_{n} ⁣ (⋃_{k \geq 1} A_{k}) = \frac{1}{n} \sum_{i = 1}^{n} \sum_{k \geq 1} 1 {x_{i} \in A_{k}} = \sum_{k \geq 1} \frac{1}{n} \sum_{i = 1}^{n} 1 {x_{i} \in A_{k}} = \sum_{k \geq 1} P_{n} (A_{k}) .$ The interchange is between a finite sum and a nonnegative series; by [F4] both are the supremum of the corresponding finite partial sums, so finite distributivity gives the equality.

2.1F2F3step 1.1step 1.2∎

Steps 1.1 and 1.2 verify the empty-set and countable-additivity clauses in [F2], and step 1.1 gives total mass one. Thus [F3] makes $P_{n}$ a probability measure.

Depends on

Used by

Cited to discharge well-definedness by IID empirical bootstrap sample and mean statistic.

Dependency tree · two levels

19 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

University of Washington STAT 581, Chapter 8: Bootstrap and Jackknife Estimation of Sampling Distributions, §2 (standard reference, not scraped)
Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §1 (standard reference, not scraped)