Alphabeta MathHow 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.
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- 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 product sigma-algebra and its finite iterates
Definition
Let and be measurable spaces. Their product sigma-algebra is
the sigma-algebra on generated by the measurable rectangles of Measurable rectangles in a product of measurable spaces.
For the empty family, put and define its empty product sigma-algebra by
For a nonempty finite family of measurable spaces with , the finite product sigma-algebra is defined recursively by
This recursive definition is the finite base case used later when the library passes to countable product constructions.
Depends on
Used by
- A consistent family of finite-dimensional distributions Definition
- Coordinate maps, finite-coordinate cylinders, and the cylinder σ-algebra Definition
- Process law, modification, and indistinguishability Definition
- Progressively measurable and predictable processes Definition
- Sections Eₓ, Eʸ, fₓ, and fʸ on a product Definition
- Closed witness codings and completion measurability of Borel projections Lemma
- Conditioning a known state and independent noise Lemma
- Finite disjoint unions of measurable rectangles form an algebra generating the product sigma-algebra Lemma
- Finite products of standard Borel spaces are standard Borel Lemma
- L² convolution on a compact group is Hilbert–Schmidt Lemma
- Product rectangle kernels are dense in product L two Lemma
- Finite product measures are the base case for countable product constructions Remark
- Every section of a product-measurable set is measurable Theorem
- Expectations factor over finite products of independent random variables Theorem
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique Theorem
- Independent random elements have product joint law Theorem
- Measurability of integration against a kernel Theorem
- Newtonian potentials solve the distributional Poisson equation Theorem
Dependency tree · two levels
6 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
- John K. Hunter, Measure Theory, Section 5.1 (standard reference, not scraped)