Mem HS iff
QuantumMechanics.OneDimension.HilbertSpace.memHS_iff
Plain-language statement
A function f satisfies MemHS f if and only if it is almost everywhere strongly measurable, and square integrable.
Exact Lean statement
lemma memHS_iff {f : ℝ → ℂ} : MemHS f ↔
AEStronglyMeasurable f ∧ Integrable (fun x => ‖f x‖ ^ 2)Formal artifact
Lean source
lemma memHS_iff {f : ℝ → ℂ} : MemHS f ↔ AEStronglyMeasurable f ∧ Integrable (fun x => ‖f x‖ ^ 2) := by rw [MemHS, MemLp, and_congr_right_iff] intro h1 rw [MeasureTheory.eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top (Ne.symm (NeZero.ne' 2)) ENNReal.ofNat_ne_top] simp only [ENNReal.toReal_ofNat, ENNReal.rpow_ofNat, Integrable] have h0 : MeasureTheory.AEStronglyMeasurable (fun x => norm (f x) ^ 2) MeasureTheory.volume := MeasureTheory.AEStronglyMeasurable.pow (continuous_norm.comp_aestronglyMeasurable h1) .. simp [h0, HasFiniteIntegral]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/HilbertSpaces/OneDimension/Basic.lean:68-77
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