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.
- 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.
in as , for
Statement
Assume the Axiom of Countable Choice.
Let and . Then
Facts & Assumptions
Given: The Axiom of Countable Choice, , , and .
is dense in ( is dense in for ).
Compactly supported continuous functions are translation-continuous in (Continuous compactly supported functions are translation-continuous in ).
Lebesgue measure is translation invariant, so (Translation of a function on , Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation).
Minkowski's inequality is available (Minkowski's inequality for integrals, including ).
Proof
By [L1], choose with [L1, L2, given, choose] . By [L2], choose such that implies .
For , [L3] and [L4] give [L3, L4, step 1.1, algebra]
Since was arbitrary, as [step 2.1] .
Depends on
- Translation of a function on $\mathbb{R}^n$
- $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ for $1 \le p < \infty$
- Continuous compactly supported functions are translation-continuous in $L^p$
- Minkowski's inequality for integrals, including $p = \infty$
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
Used by
- If 1 < p < ∞ and q is conjugate to p, then f*g is continuous and vanishes at infinity Corollary
- The tightness hypothesis of the Fr'echet--Kolmogorov criterion cannot be dropped Counterexample
- Affine flux reduces the entropy semigroup to translation Example
- The right-translation semigroup on Lp has the weak derivative as generator Example
- FALSE: translation is continuous in L^∞ False statement
- Compactly supported smooth functions are dense in Slobodeckij spaces Lemma
- Complex translation, convolution, approximate identities, and mollification Lemma
- Relative compactness forces uniform translation continuity in Lᵖ Lemma
- The Kruzhkov doubling inequality for two entropy solutions Lemma
- Every L¹ approximate identity converges to the identity in Lᵖ for 1 ≤ p < ∞ Theorem
- Riemann–Lebesgue lemma Theorem
- The difference-quotient characterisation of W^1,p for 1<p<∞ Theorem
- The kernel of the trace is the closure of the test functions Theorem
- The principle of descent and the logarithmic domination principle Theorem
Dependency tree · two levels
31 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
- Terence Tao, An Introduction to Measure Theory (standard reference, not scraped)
- Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)