Learning.measurable_algorithm_iff
No docstring.
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๐ : Type u_1m๐ : MeasurableSpace ๐A measurable space is a space equipped with a ฯ-algebra. -
๐ : Type u_2m๐ : MeasurableSpace ๐ -
๐จ : Type u_3m๐จ : MeasurableSpace ๐จ -
ฮฉ : Type u_4mฮฉ : MeasurableSpace ฮฉ
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f : ฮฉ โ Algorithm ๐ ๐ ๐จA stochastic, sequential algorithm.
Measurable f โ โ (n : โ), Measurable fun x => (f x).policy nMeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
Learning.Algorithm : (๐ : Type u_5) โ
(๐ : Type u_6) โ
(๐จ : Type u_7) โ [MeasurableSpace ๐] โ [MeasurableSpace ๐] โ [MeasurableSpace ๐จ] โ Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `๐`, then takes an action in `๐`, and finally receives feedback in `๐จ`. The action is a random function of the past rounds and the current observation.Go to its page
Iff : Prop โ Prop โ PropIf and only if, or logical bi-implication. `a โ b` means that `a` implies `b` and vice versa. By `propext`, this implies that `a` and `b` are equal and hence any expression involving `a` is equivalent to the corresponding expression with `b` instead. Conventions for notations in identifiers: * The recommended spelling of `โ` in identifiers is `iff`. * The recommended spelling of `<->` in identifiers is `iff` (prefer `โ` over `<->`).
Measurable : {ฮฑ : Type u_1} โ {ฮฒ : Type u_2} โ [MeasurableSpace ฮฑ] โ [MeasurableSpace ฮฒ] โ (ฮฑ โ ฮฒ) โ PropA function `f` between measurable spaces is measurable if the preimage of every measurable set is measurable.
Nat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
Code
lemma measurable_algorithm_iff (f : ฮฉ โ Algorithm ๐ ๐ ๐จ) :
Measurable f โ โ n, Measurable fun x โฆ (f x).policy nProof
by unfold instMeasurableSpaceAlgorithm rw [measurable_comap_iff, measurable_pi_iff] simp
New in v4.34.0-rc2-78-g3ff93c0 (2026-09-10), and its meaning has not changed since.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file ยท open it in the Lean web editor.
Dependency graph
Audit surface: 5 project declarations, 16 external constants
โ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.