False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-21

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FALSE: continuity from above needs no finiteness hypothesis

Statement

False claim. For every decreasing sequence $(E_{k})$ of measurable sets, one has $μ (⋂_{k} E_{k}) = \inf_{k} μ (E_{k})$ , without requiring any $E_{k}$ to have finite measure. The valid theorem Continuity from above when one set has finite measure includes precisely that missing hypothesis.

Facts & Assumptions

Given: Counting measure $#$ on $N$ and the tails $E_{k} : = {n \in N : k \leq n}$ .

[L1]

Counting measure assigns $+ \infty$ to every infinite set (Counting measure on an arbitrary set) and is a measure (Counting measure is a measure).

[L2]

Continuity from above is proved when one member of the decreasing sequence has finite measure (Continuity from above when one set has finite measure).

[L3]

The natural order is defined by $m \leq n$ exactly when $m + k = n$ for some natural $k$ (Order on the natural numbers), natural addition is cancellative (Addition is cancellative), and $k < k + 1$ (Discreteness: $σ (n)$ is the immediate successor).

Refutation

technique · direct

1.1givenL3

The tails decrease, $E_{k + 1} \subseteq E_{k}$ , and $E_{0} = N$ .

1.2givenL1L3

Every $E_{k}$ is infinite, because $n \mapsto k + n$ injects $N$ into it; hence $# (E_{k}) = + \infty$ for every $k$ .

1.3givenL3

The intersection is empty: if $n$ belonged to every tail, it would belong to $E_{n + 1}$ , which would say $n + 1 \leq n$ , contrary to discreteness of the natural order.

2.1step 1.2step 1.3L1L2∎

Thus $# (⋂_{k} E_{k}) = # (\emptyset) = 0$ but $\inf_{k} # (E_{k}) = + \infty$ . This refutes the claim and shows why [L2] cannot be applied: no tail has finite counting measure.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

G. Folland, Real Analysis, 2nd ed., Theorem 1.8(d) (standard reference, not scraped)