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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A small set of reals is both null and meagre under MA
Statement
Under MA, if is a set of reals with , then is both Lebesgue null and meagre, though these notions are independent for arbitrary sets.
Facts & Assumptions
Given: The hypotheses, objects, and conventions in the Statement.
MA makes unions of fewer than continuum many meagre sets meagre handles small unions of meagre sets.
MA makes unions of fewer than continuum many null sets null handles small unions of null sets.
Proof
Write . Every singleton is closed nowhere dense and has Lebesgue measure zero.
Since the index family has size below the continuum, F1 makes its union meagre and F2 makes it null. The two conclusions are proved separately; neither is inferred from the other.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
9 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
- Kunen, Set Theory, consequences of Martin's Axiom (standard reference, not scraped)