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.

Sets whose symmetric difference is null have the same measure

Statement

If $A$ and $B$ are measurable and $μ (A △ B) = 0$ , where $A △ B = (A ∖ B) \cup (B ∖ A)$ , then $μ (A) = μ (B)$ .

Facts & Assumptions

Given: Measurable sets $A, B$ with $μ (A △ B) = 0$ .

[L1]

Measures are monotone: $C \subseteq D$ implies $μ (C) \leq μ (D)$ (Measures are monotone).

[L2]

For measurable $C, D$ , $μ (C \cup D) + μ (C \cap D) = μ (C) + μ (D)$ (The two-set measure identity $μ (A \cup B) + μ (A \cap B) = μ (A) + μ (B)$ ).

Proof

technique · direct

1.1givenL1

Monotonicity gives $μ (A ∖ B) = μ (B ∖ A) = 0$ , since both sets lie in $A △ B$ .

2.1step 1.1L2

Applying the two-set identity to $A \cap B$ and $A ∖ B$ , which are disjoint and have union $A$ , gives $μ (A) = μ (A \cap B) + 0$ ; the same argument gives $μ (B) = μ (A \cap B) + 0$ .

3.1step 2.1∎

Hence $μ (A) = μ (B)$ , including when the common value $μ (A \cap B)$ is $+ \infty$ .

Depends on

Dependency tree · 2 levels

Sources

G. Folland, Real Analysis, 2nd ed., §1.3, Exercise 12(a)