File: Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean

full_name: ae_eq_zero_of_integral_smooth_smul_eq_zero
code: /-- If a locally integrable function `f` on a finite-dimensional real manifold has zero integral
when multiplied by any smooth compactly supported function, then `f` vanishes almost everywhere. -/
theorem ae_eq_zero_of_integral_smooth_smul_eq_zero (hf : LocallyIntegrable f μ)
    (h : ∀ g : M → ℝ, Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, f x = 0 
goal: ∀ᵐ (x : M) ∂μ, f x = 0

full_name: IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero'
code: /-- If a function `f` locally integrable on an open subset `U` of a finite-dimensional real
  manifold has zero integral when multiplied by any smooth function compactly supported
  in `U`, then `f` vanishes almost everywhere in `U`. -/
nonrec theorem IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero' {U : Set M} (hU : IsOpen U)
    (hSig : IsSigmaCompact U) (hf : LocallyIntegrableOn f U μ)
    (h : ∀ g : M → ℝ,
      Smooth I 𝓘(ℝ) g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, x ∈ U → f x = 0 
goal: ∀ᵐ (x : M) ∂μ, x ∈ U → f x = 0

full_name: IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero
code: theorem IsOpen.ae_eq_zero_of_integral_smooth_smul_eq_zero {U : Set M} (hU : IsOpen U)
    (hf : LocallyIntegrableOn f U μ)
    (h : ∀ g : M → ℝ,
      Smooth I 𝓘(ℝ) g → HasCompactSupport g → tsupport g ⊆ U → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, x ∈ U → f x = 0 
goal: ∀ᵐ (x : M) ∂μ, x ∈ U → f x = 0

full_name: ae_eq_of_integral_smooth_smul_eq
code: /-- If two locally integrable functions on a finite-dimensional real manifold have the same integral
when multiplied by any smooth compactly supported function, then they coincide almost everywhere. -/
theorem ae_eq_of_integral_smooth_smul_eq
    (hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ),
      Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :
    ∀ᵐ x ∂μ, f x = f' x 
goal: ∀ᵐ (x : M) ∂μ, f x = f' x

full_name: ae_eq_zero_of_integral_contDiff_smul_eq_zero
code: /-- If a locally integrable function `f` on a finite-dimensional real vector space has zero integral
when multiplied by any smooth compactly supported function, then `f` vanishes almost everywhere. -/
theorem ae_eq_zero_of_integral_contDiff_smul_eq_zero (hf : LocallyIntegrable f μ)
    (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, f x = 0 
goal: ∀ᵐ (x : E) ∂μ, f x = 0

full_name: ae_eq_of_integral_contDiff_smul_eq
code: /-- If two locally integrable functions on a finite-dimensional real vector space have the same
integral when multiplied by any smooth compactly supported function, then they coincide almost
everywhere. -/
theorem ae_eq_of_integral_contDiff_smul_eq
    (hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : E → ℝ),
      ContDiff ℝ ⊤ g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :
    ∀ᵐ x ∂μ, f x = f' x 
goal: ∀ᵐ (x : E) ∂μ, f x = f' x

full_name: IsOpen.ae_eq_zero_of_integral_contDiff_smul_eq_zero
code: /-- If a function `f` locally integrable on an open subset `U` of a finite-dimensional real
  manifold has zero integral when multiplied by any smooth function compactly supported
  in an open set `U`, then `f` vanishes almost everywhere in `U`. -/
theorem IsOpen.ae_eq_zero_of_integral_contDiff_smul_eq_zero {U : Set E} (hU : IsOpen U)
    (hf : LocallyIntegrableOn f U μ)
    (h : ∀ (g : E → ℝ), ContDiff ℝ ⊤ g → HasCompactSupport g → tsupport g ⊆ U →
        ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, x ∈ U → f x = 0 
goal: ∀ᵐ (x : E) ∂μ, x ∈ U → f x = 0