ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A countable probability space with geometric weights

Example

Let $Ω = N$ and define $P (A) : = \sum_{n \in A} 2^{- (n + 1)} (A \subseteq N) .$ Let $X : Ω \to R$ be the coordinate map $X (n) = n$ . Then $X$ is a random variable, its law is $P_{X} ({n}) = 2^{- (n + 1)},$ and $E [X] = 1.$

Facts & Assumptions

Given: The weights $2^{- (n + 1)}$ on $N$ and the coordinate map $X (n) = n$ .

[L1]

A probability measure has total mass $1$ , the law of a random element is its pushforward measure, and change of variables computes expectation from the law (Probability measures and probability spaces, Law or distribution of a random element, Change of variables for expectation).

[L2]

A real random variable is a measurable map into $R$ (Random elements and real random variables).

Verification

technique · direct

1.1

The geometric series gives $P (N) = \sum_{n = 0}^{\infty} 2^{- (n + 1)} = 1,$ so the displayed weights define a probability measure as in [L1]. Since every subset of $N$ is measurable, the coordinate map $X$ is a real random variable by [L2].

L1L2algebra

1.2

For each $n \in N$ , the fibre of ${n}$ is the singleton ${n}$ , so the law definition gives $P_{X} ({n}) = P ({n}) = 2^{- (n + 1)} .$

2.1

Applying change of variables from [L1] to the identity function on $N \subseteq R$ gives $E [X] = \sum_{n = 0}^{\infty} n 2^{- (n + 1)} = 1.$

step 1.2L1algebra

3.1

Steps 1.1, 1.2, and 2.1 verify the measure, the law, and the expectation.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.1 (standard reference, not scraped)