E Lp Norm indicator tail eq set Integral of nonneg
MeasureTheory.eLpNorm_indicator_tail_eq_setIntegral_of_nonneg
Project documentation
A helper lemma for uniformIntegrable_iff_tendsto_iSup_setIntegral_of_nonneg.
Exact Lean statement
lemma eLpNorm_indicator_tail_eq_setIntegral_of_nonneg {f : Ω → ℝ}
(hf : Integrable f μ) (hnonneg : 0 ≤ᵐ[μ] f) (c : ℝ≥0) :
eLpNorm ({ω | c < ‖f ω‖₊}.indicator f) 1 μ =
ENNReal.ofReal (∫ ω in {ω | c < f ω}, f ω ∂μ)Formal artifact
Lean source
lemma eLpNorm_indicator_tail_eq_setIntegral_of_nonneg {f : Ω → ℝ} (hf : Integrable f μ) (hnonneg : 0 ≤ᵐ[μ] f) (c : ℝ≥0) : eLpNorm ({ω | c < ‖f ω‖₊}.indicator f) 1 μ = ENNReal.ofReal (∫ ω in {ω | c < f ω}, f ω ∂μ) := by rw [eLpNorm_indicator_tail_eq_setIntegral_norm hf] apply congrArg calc _ = ∫ (ω : Ω) in {ω | c < f ω}, ‖f ω‖ ∂μ := by apply setIntegral_congr_set filter_upwards [hnonneg] with ω hω simp [Set.ofPred, abs_of_nonneg hω] _ = _ := by refine setIntegral_congr_fun₀ (aestronglyMeasurable_const.nullMeasurableSet_lt hf.aestronglyMeasurable) fun ω hω => ?_ exact Real.norm_of_nonneg (c.2.trans hω.le)- Project
- Brownian motion
- License
- Apache-2.0
- Commit
- 5077304f5e73
- Source
- BrownianMotion/StochasticIntegral/UniformIntegrable.lean:99-113
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Related declarations
Continuous Within At Iio indicator Ioc
continuousWithinAt_Iio_indicator_Ioc
Plain-language statement
The indicator of a half-open interval Ioc a b with constant value c is left-continuous: when approached from the left it is eventually constant, so it is continuous within Iio t at t for every t.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Iio ne Bot
Dense.comap_val_nhdsWithin_Iio_neBot
Plain-language statement
This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.
Source project: Brownian motion
Person-level attribution pending.
Dense comap val nhds Within Ioi ne Bot
Dense.comap_val_nhdsWithin_Ioi_neBot
Project documentation
This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.
Source project: Brownian motion
Person-level attribution pending.