Theorems · Theorem · functional analysis
MeasureTheory.MemLp.exist_eLpNorm_sub_le
∀ {E : Type u_3} {F : Type u_4} [inst : MeasurableSpace E] [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E]
[inst_6 : NormedSpace ℝ F] {μ : MeasureTheory.Measure E} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {p : ENNReal},
p ≠ ⊤ →
1 ≤ p →
∀ {f : E → F},
MeasureTheory.MemLp f p μ →
∀ {ε : ℝ},
0 < ε → ∃ g, HasCompactSupport g ∧ ContDiff ℝ (↑⊤) g ∧ MeasureTheory.eLpNorm (f - g) p μ ≤ ENNReal.ofReal εEvery Lp function can be approximated by a smooth compactly supported function provided that
p < ∞.
- Defined in
- Mathlib.Analysis.Normed.Lp.SmoothApprox
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 304 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- MeasurableSpacestatement and proof · cited by 13,106
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- Top.topstatement and proof · cited by 9,680
- ENatstatement · cited by 4,985
- Continuousproof · cited by 2,592
- LT.lt.leproof · cited by 2,189
- le_reflproof · cited by 2,061
- FiniteDimensionalstatement and proof · cited by 1,854
Cited by2
Results whose statement or proof uses this declaration.
- SchwartzMap.denseRange_toLpCLMproof · cited by 3
- MeasureTheory.Lp.dense_hasCompactSupport_contDiffproof · cited by 0