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The lower integral

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

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    Natural Language Statement

    Definition

    For a measure μ\mu and a function g:Ω→Rg : \Omega \to \mathbb{R}, the lower integral is

    Lμ(g)=inf⁡M∈N∫max⁡(g,−M) dμ ∈[−∞,∞],L_\mu(g) = \inf_{M \in \mathbb{N}} \int \max(g, -M)\, d\mu \ \in [-\infty, \infty],

    the decreasing limit of the integrals of the truncations of gg from below. When g+g^+ is integrable every truncation is integrable, and Lμ(g)∈[−∞,∞)L_\mu(g) \in [-\infty, \infty) is the integral of gg, allowing the value −∞-\infty; when gg is integrable it is the ordinary integral.

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    Lean source view on GitLab

    1import Mathlib.MeasureTheory.Integral.Bochner.Basic
    2import Mathlib.Data.EReal.Operations
    3
    4/-!
    5---
    6title: The lower integral
    7type: definition
    8---
    9For a measure μ\mu and a function g:Ω→Rg : \Omega \to \mathbb{R}, the lower integral is
    10Lμ(g)=inf⁡M∈N∫max⁡(g,−M) dμ ∈[−∞,∞],L_\mu(g) = \inf_{M \in \mathbb{N}} \int \max(g, -M)\, d\mu \ \in [-\infty, \infty],
    11the decreasing limit of the integrals of the truncations of gg from below. When g+g^+ is
    12integrable every truncation is integrable, and Lμ(g)∈[−∞,∞)L_\mu(g) \in [-\infty, \infty) is the integral of
    13gg, allowing the value −∞-\infty; when gg is integrable it is the ordinary integral.
    14-/
    15
    16namespace Lax606786.LowerIntegrals
    17
    18open MeasureTheory
    19
    20/-- `L_μ(g) = inf_M ∫ max(g, -M) dμ`. -/
    21noncomputable def lowerIntegral {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (g : Ω → ℝ) :
    22 EReal :=
    23 ⨅ M : ℕ, ((∫ ω, max (g ω) (-(M : ℝ)) ∂μ : ℝ) : EReal)
    24
    25end Lax606786.LowerIntegrals
    26
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