PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-21

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.

Measures are monotone

Statement

Let $μ$ be a measure on $(X, A)$ . If $A, B \in A$ and $A \subseteq B$ , then $μ (A) \leq μ (B)$ .

Facts & Assumptions

Given: A measure $μ$ on $(X, A)$ and measurable sets $A \subseteq B$ .

[L1]

A measure is nonnegative and countably additive on pairwise disjoint measurable sequences (Measures on sigma-algebras).

Proof

technique · direct

1.1given

The sets $A$ and $B ∖ A$ are measurable, disjoint, and have union $B$ .

2.1step 1.1L1algebra∎

Countable additivity applied to these two sets and empty sets thereafter gives $μ (B) = μ (A) + μ (B ∖ A) \geq μ (A)$ ; no subtraction is used, so the argument also covers $μ (B) = + \infty$ and the degenerate cases $A = \emptyset$ and $A = B$ .

Measures are monotone

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources