Learning.Algorithm.AbsolutelyContinuous.p0
No docstring.
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𝓞 : Type u_1MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra. -
𝓐 : Type u_2MeasurableSpace 𝓐 -
𝓨 : Type u_3MeasurableSpace 𝓨
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alg : Algorithm 𝓞 𝓐 𝓨A stochastic, sequential algorithm. -
alg₀ : Algorithm 𝓞 𝓐 𝓨 -
o : 𝓞
(alg.p0 o).AbsolutelyContinuous (alg₀.p0 o)μ is absolutely continuous with respect to ν, or that μ is dominated by ν, if ν(A) = 0 implies that μ(A) = 0.MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Learning.Algorithm : (𝓞 : Type u_5) →
(𝓐 : Type u_6) →
(𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives feedback in `𝓨`. The action is a random function of the past rounds and the current observation.Go to its page
Learning.Algorithm.AbsolutelyContinuous : {𝓞 : Type u_1} →
{𝓐 : Type u_2} →
{𝓨 : Type u_3} →
[inst : MeasurableSpace 𝓞] →
[inst_1 : MeasurableSpace 𝓐] →
[inst_2 : MeasurableSpace 𝓨] → Learning.Algorithm 𝓞 𝓐 𝓨 → Learning.Algorithm 𝓞 𝓐 𝓨 → PropFor every time and history, the distribution over actions according to `alg` is absolutely continuous with respect to the distribution over actions according to `alg₀`.Go to its page
MeasureTheory.Measure.AbsolutelyContinuous : {α : Type u_1} → {_m0 : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → PropWe say that `μ` is absolutely continuous with respect to `ν`, or that `μ` is dominated by `ν`, if `ν(A) = 0` implies that `μ(A) = 0`.
Learning.Algorithm.p0 : {𝓞 : Type u_1} →
{𝓐 : Type u_2} →
{𝓨 : Type u_3} →
{m𝓞 : MeasurableSpace 𝓞} →
{m𝓐 : MeasurableSpace 𝓐} → {m𝓨 : MeasurableSpace 𝓨} → Learning.Algorithm 𝓞 𝓐 𝓨 → ProbabilityTheory.Kernel 𝓞 𝓐Distribution of the first action given the first observation: the policy at time `0` applied to the empty history.Go to its page
Code
lemma AbsolutelyContinuous.p0 {alg alg₀ : Algorithm 𝓞 𝓐 𝓨} (h : alg ≪ₐ alg₀) (o : 𝓞) :
alg.p0 o ≪ alg₀.p0 oProof
h.policy 0 (default, o)
Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10), the 2th recorded change.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Audit surface: 5 project declarations, 18 external constants
✓ 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.