RemarkRemark: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-27

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.

The convention $0 \cdot \infty = 0$ is used only for pointwise products of measurable functions

On this page the symbol $f g$ for two $\overline{R}$ -valued measurable functions is a pointwise definition, not a limit rule. At points where one factor is $0$ and the other is infinite, we set

$0 \cdot (+ \infty) = 0 \cdot (- \infty) = 0.$

This convention exists only so that decompositions such as $f = f^{+} - f^{-}$ and the measurable-product theorem stay pointwise total on the extended reals. It does not contradict Null times divergent has no rule: $x_{k} = 1 / k$ with $y_{k} = c k$ gives product limit $c$ , and with $y_{k} = k^{2}$ gives divergence, because that counterexample concerns limits of ordinary real products, not the value of a pointwise formula.

Depends on

The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$, its order, and the arithmetic that is left undefined

Used by

Arithmetic and lattice operations preserve measurability whenever they are defined Theorem

Dependency tree · two levels

7 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

Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)
John K. Hunter, Measure Theory, Definition 3.3 (standard reference, not scraped)

The convention 0⋅∞=00⋅∞=0 is used only for pointwise products of measurable functions

Depends on

Used by

Dependency tree · two levels

Sources

The convention $0 \cdot \infty = 0$ is used only for pointwise products of measurable functions