LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Indicators turn event probabilities, intersections, and finite counts into expectations and products

Statement

For every event $A$ , $E [1_{A}] = P (A)$ . For every finite family $(A_{i})_{i \in I}$ , $\prod_{i \in I} 1_{A_{i}} = 1_{\cap_{i \in I} A_{i}},$ and $\sum_{i \in I} 1_{A_{i}} (ω)$ is the number of events $A_{i}$ that contain $ω$ . The empty product is $1$ and the empty sum is $0$ .

Facts & Assumptions

Given: Events $A$ and $(A_{i})_{i \in I}$ in one finite probability space.

[L1]

The indicator of an event is $1$ on the event and $0$ off it (The indicator random variable of an event).

[L2]

Expectation is the finite weighted sum over outcomes (Expectation of a real random variable on a finite probability space).

[L3]

Empty finite sums and products are $0$ and $1$ (Finite sums and finite products, by recursion).

Proof

technique · direct

1.1

Expanding $E [1_{A}]$ leaves exactly the weights of outcomes in $A$ , hence equals $P (A)$ .

L1L2

1.2

At an outcome $ω$ , the product $\prod_{i} 1_{A_{i}} (ω)$ is $1$ exactly when $ω$ belongs to every $A_{i}$ , and is otherwise $0$ .

1.3

At $ω$ , each summand $1_{A_{i}} (ω)$ contributes one exactly when $ω \in A_{i}$ , so their sum counts those events.

2.1

Steps 1.2 and 1.3 also give the stated empty conventions by [L3].

step 1.1step 1.2step 1.3L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Lemma 3.1.3 (standard reference, not scraped)
M. Bucic, Probabilistic Method, Section 2.1 (standard reference, not scraped)