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A finite empirical measure is a probability law

Statement

Let $(X, A)$ be a measurable space, let $n \geq 1$ , and let $x_{1}, \dots, x_{n} \in X$ . Define

$P_{n} (A) : = \frac{1}{n} \sum_{i = 1}^{n} δ_{x_{i}} (A), A \in A .$

Then $P_{n}$ is a probability measure on $(X, A)$ .

Facts & Assumptions

Given: A measurable space $(X, A)$ , an integer $n \geq 1$ , points $x_{1}, \dots, x_{n} \in X$ , and the displayed set function $P_{n}$ .

[F1]

The Dirac set function at a point: $δ_{x} (A) = 1$ when $x \in A$ and $0$ otherwise.

[F2]

Probability measures and probability spaces: a probability measure is a countably additive measure of total mass one.

Proof

technique · direct

1.1givenF1algebra

Each summand in $P_{n} (A)$ is nonnegative, $P_{n} (\emptyset) = 0$ , and $P_{n} (X) = n^{- 1} \sum_{i = 1}^{n} 1 = 1$ by [F1].

1.2givenF1algebra

Let $(A_{j})_{j \geq 1}$ be pairwise disjoint. For each $j$ , let $I_{j} : = {i \in {1, \dots, n} : x_{i} \in A_{j}}$ . The sets $I_{j}$ are pairwise disjoint, and at most $n$ of them are nonempty, because each index $i$ belongs to at most one $I_{j}$ . Their union is ${i : x_{i} \in ⋃_{j} A_{j}}$ . Hence the number of sample indices in $⋃_{j} A_{j}$ is $\sum_{j} # I_{j}$ , and dividing by $n$ gives $P_{n} (⋃_{j} A_{j}) = \sum_{j} P_{n} (A_{j})$ .

2.1step 1.1step 1.2F2∎

Steps 1.1–1.2 show that $P_{n}$ is a nonnegative countably additive set function with total mass one; by [F2] it is a probability measure.

Depends on

Used by

Bounded influence can coexist with vanishing replacement breakdown Counterexample
m-out-of-n bootstrap and subsampling Definition
The lower median of an odd empirical sample is its middle order statistic Lemma

Dependency tree · two levels

5 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

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §1.3, PDF pp. 13–14 (standard reference, not scraped)
Elvezio Ronchetti, Introduction to Robust Statistics, slides 15 and 18 (standard reference, not scraped)