`Learning.IsAlgEnvSeqUntil.measurable_history`🔗

Lemma

No docstring.

Types

𝓞 : Type u_1m𝓞 : MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra.
𝓐 : Type u_2m𝓐 : MeasurableSpace 𝓐
𝓨 : Type u_3m𝓨 : MeasurableSpace 𝓨
Ω : Type u_4mΩ : MeasurableSpace Ω

Given

O : ℕ → Ω → 𝓞
A : ℕ → Ω → 𝓐
Y : ℕ → Ω → 𝓨
alg : Algorithm 𝓞 𝓐 𝓨A stochastic, sequential algorithm.
env : Environment 𝓞 𝓐 𝓨A stochastic environment.
P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsFiniteMeasure PA measure μ is called finite if μ univ < ∞.
N : ℕ
n : ℕ

Assuming

h : IsAlgEnvSeqUntil O A Y alg env P NAn algorithm-environment sequence until time N: a sequence of observations, actions and feedbacks such that the rounds at times n < N are generated by an algorithm interacting with an environment.

Then

Measurable (history O A Y n)

A function f between measurable spaces is measurable if the preimage of every measurable set is measurable.

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

Nat : Type

The 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.

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.Environment : (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 environment.
At each round, an observation is drawn prior to the algorithm taking an action. Then the environment
provides feedback based on the observation and the action.

Go to its page

MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called finite if `μ univ < ∞`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Learning.IsAlgEnvSeqUntil : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {Ω : Type u_4} →
        {m𝓞 : MeasurableSpace 𝓞} →
          {m𝓐 : MeasurableSpace 𝓐} →
            {m𝓨 : MeasurableSpace 𝓨} →
              {mΩ : MeasurableSpace Ω} →
                (ℕ → Ω → 𝓞) →
                  (ℕ → Ω → 𝓐) →
                    (ℕ → Ω → 𝓨) →
                      Learning.Algorithm 𝓞 𝓐 𝓨 →…

An algorithm-environment sequence until time `N`: a sequence of observations, actions and
feedbacks such that the rounds at times `n < N` are generated by an algorithm interacting with
an environment. In particular, the law of `history O A Y N` is determined.

Go to its page

Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable.

Learning.history : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} → {Ω : Type u_4} → (ℕ → Ω → 𝓞) → (ℕ → Ω → 𝓐) → (ℕ → Ω → 𝓨) → (n : ℕ) → Ω → Learning.Hist 𝓞 𝓐 𝓨 n

History of the algorithm-environment sequence before time `n`: the rounds at
times `0, ..., n - 1`.

Go to its page

Code

lemma IsAlgEnvSeqUntil.measurable_history (h : IsAlgEnvSeqUntil O A Y alg env P N) (n : ℕ) :
    Measurable (history O A Y n)

Proof

by
  have hO := h.measurable_obs
  have hA := h.measurable_action
  have hY := h.measurable_feedback
  fun_prop

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: 6 project declarations, 25 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.

Learning.IsAlgEnvSeqUntil.measurable_history🔗

`Learning.IsAlgEnvSeqUntil.measurable_history`🔗