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Properties of the lower integral

Lax606786.LowerIntegralProperties · concepts/Lax606786/LowerIntegralProperties.lean · lax-606786

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    Natural Language Statement

    Theorem

    For every measure μ\mu and g:Ω→Rg : \Omega \to \mathbb{R}, Lμ(g)<∞L_\mu(g) < \infty. If μ\mu is a probability measure and gg is integrable, then Lμ(g)=∫g dμL_\mu(g) = \int g \, d\mu.

    Concept map
    2 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 lowerIntegral_eq_integral proven

    2 lowerIntegral_lt_top proven

    Lean source view on GitLab

    1import Lax606786.LowerIntegrals
    2
    3/-!
    4---
    5title: Properties of the lower integral
    6type: theorem
    7---
    8For every measure μ\mu and g:Ω→Rg : \Omega \to \mathbb{R}, Lμ(g)<∞L_\mu(g) < \infty. If μ\mu is a
    9probability measure and gg is integrable, then Lμ(g)=∫g dμL_\mu(g) = \int g \, d\mu.
    10-/
    11
    12namespace Lax606786.LowerIntegralProperties
    13
    14open MeasureTheory
    15open Lax606786.LowerIntegrals
    16
    17variable {Ω : Type*} [MeasurableSpace Ω]
    18
    19/-- `L_μ(g) < ∞`. -/
    20axiom lowerIntegral_lt_top (μ : Measure Ω) (g : Ω → ℝ) : lowerIntegral μ g < ⊤
    21
    22/-- On integrable functions the lower integral is the integral. -/
    23axiom lowerIntegral_eq_integral {μ : Measure Ω} [IsProbabilityMeasure μ] {g : Ω → ℝ}
    24 (hg : Integrable g μ) : lowerIntegral μ g = ((∫ ω, g ω ∂μ : ℝ) : EReal)
    25
    26end Lax606786.LowerIntegralProperties
    27
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