LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

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

Basic identities for a probability measure

Statement

Let $(Ω, F, P)$ be a probability space, let $A, B \in F$ , and let $(A_{n})_{n \in N}$ be events.

$P (A^{c}) = 1 - P (A)$ .
If $A \subseteq B$ , then $P (A) \leq P (B)$ .
If $A \subseteq B$ , then $P (B ∖ A) = P (B) - P (A)$ .
$P (⋃_{n} A_{n}) \leq \sum_{n = 0}^{\infty} P (A_{n})$ , and for every natural $m$ , $P (⋃_{n < m} A_{n}) \leq \sum_{n < m} P (A_{n}) .$
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$ .
If $A_{n} ↑ A$ , then $P (A) = \sup_{n} P (A_{n}) .$ If $A_{n} ↓ A$ , then $P (A) = \inf_{n} P (A_{n}) .$

Facts & Assumptions

Given: A probability space $(Ω, F, P)$ , events $A, B$ , and an event sequence $(A_{n})$ .

[L1]

A probability measure is a measure of total mass $1$ (Probability measures and probability spaces).

[L2]

Measures are monotone, set differences subtract when the smaller set has finite measure, subadditivity holds, continuity from below holds, continuity from above holds once one set has finite measure, and finite inclusion-exclusion holds for finite-measure sets (Measures are monotone, Measure of a set difference when the smaller set has finite measure, Finite and countable subadditivity of measures, Continuity from below for measures, Continuity from above when one set has finite measure, Inclusion-exclusion for a nonempty finite family of finite-measure sets).

Proof

technique · direct

1.1

Because $P (Ω) = 1$ by [L1] and $A \subseteq Ω$ , [L2] gives $1 = P (Ω) = P (A) + P (A^{c}),$ so $P (A^{c}) = 1 - P (A)$ . Monotonicity in [L2] also gives $A \subseteq B \Rightarrow P (A) \leq P (B)$ .

L1L2

1.2

Since every probability is at most $1$ , the finite-measure hypotheses in [L2] apply to $A$ , $B$ , and $A \cup B$ . Thus if $A \subseteq B$ , then $P (B ∖ A) = P (B) - P (A),$ subadditivity gives the countable and finite union bounds, finite inclusion-exclusion gives $P (A \cup B) = P (A) + P (B) - P (A \cap B),$ and continuity from below and from above give the two monotone-limit formulas, because a decreasing probability sequence always has finite first term.

L1L2

2.1

Steps 1.1 and 1.2 are exactly the stated probability identities.

step 1.1step 1.2∎

Depends on

Used by

The expectation of an indicator is the probability of the event Corollary

Dependency tree · two levels

17 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)
J. R. Norris, Probability and Measure, Section 1.9 (standard reference, not scraped)