Plain-English statement
The squared process norm on probability times time equals the expectation of the squared time integral.
Mathematical statement
Integral eta(t,omega)^2 d(P tensor m_[0,T]) equals integral over omega of the integral over t of eta(t,omega)^2.
Intuition
Tonelli is the rigorous bridge between the product-space L2 norm and the expected accumulated energy used by the Ito isometry.
Conditions
- finite terminal-time measure
- joint a.e. measurability of the squared process
Why these conditions cannot be dropped
- finiteness of the time measure supplies the product-measure API
- measurability is required by Tonelli
- ENNReal avoids hiding an infinite energy
Proof route
- construct the finite NNReal time measure
- form the probability-time product measure
- apply Mathlib lintegral_prod
Lean interface notes
- processL2Energy is ENNReal-valued
- the theorem does not claim the Ito isometry or L2 completion
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Lean statement
theorem chewi_display_1_1_7
{Omega : Type*} [MeasurableSpace Omega]
(eta : ℝ≥0 → Omega → ℝ) (mu : Measure Omega) (T : ℝ≥0)
(hη : AEMeasurable
(fun z : Omega × ℝ≥0 => ENNReal.ofReal ((eta z.2 z.1) ^ 2))
(processTimeMeasure mu T)) :
processL2Energy eta mu T =
∫⁻ omega, ∫⁻ t, ENNReal.ofReal ((eta t omega) ^ 2)
∂(TimeMeasure.upTo T) ∂mu :=
lintegral_prod _ hη
/-- Almost-sure local square integrability on `[0,T]`, the weaker condition
used when Chewi extends stochastic integration by localization. -/Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:94published source at 7bcd37294df1