False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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FALSE: the union of an increasing sequence of sigma-algebras is a sigma-algebra

Statement

The union of every increasing sequence of sigma-algebras on one ambient set is a sigma-algebra.

Facts & Assumptions

Given: The ambient set $X : = N$ , the tail $T_{n} : = {k \in N : k \geq n}$ , and the partition $P_{n} : = {{0}, \dots, {n - 1}, T_{n}}$ , with $P_{0} = {N}$ .

[L1]

A sigma-algebra is closed under complements and countable unions (Sigma-algebras).

Refutation

technique · counterexample

1.1L1algebra

Let $A_{n}$ be the family of unions of blocks of $P_{n}$ . Complements and countable unions correspond to complements and unions of block-index sets, so each $A_{n}$ is a sigma-algebra; splitting $T_{n}$ into ${n}$ and $T_{n + 1}$ gives $A_{n} \subseteq A_{n + 1}$ .

2.1step 1.1algebra

A member of some $A_{n}$ is finite if it omits $T_{n}$ and cofinite if it contains $T_{n}$ . Conversely every finite or cofinite subset belongs to some $A_{n}$ . Hence $⋃_{n} A_{n}$ is the finite-cofinite algebra.

3.1step 2.1L1∎

Every singleton ${2 j}$ lies in the union, but their countable union is the set of even naturals, which is neither finite nor cofinite and so is absent by step 2.1. This violates [L1].

Depends on

Sigma-algebras

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

R. F. Bass, Real Analysis for Graduate Students, version 5.0, Exercise 2.3 (standard reference, not scraped)