Theorem: MeasureTheory.not_tendsto_diracProba_of_not_tendsto
File: Mathlib/MeasureTheory/Measure/DiracProba.lean

tactic: obtain ⟨U, U_nhd, hU⟩ : ∃ U, U ∈ 𝓝 x ∧ ∃ᶠ x in L, x ∉ U := by
  by_contra! con
  apply h
  intro U U_nhd
  simpa only [not_frequently, not_not] using con U U_nhd
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))
state_after: case intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))

tactic: have Uint_nhd : interior U ∈ 𝓝 x := by simpa only [interior_mem_nhds] using U_nhd
state_before: case intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))
state_after: case intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))

tactic: obtain ⟨f, fx_eq_one, f_vanishes_outside⟩ :=
  CompletelyRegularSpace.exists_BCNN isOpen_interior.isClosed_compl
    (by simpa only [mem_compl_iff, not_not] using mem_of_mem_nhds Uint_nhd)
state_before: case intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))
state_after: case intro.intro.intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))

tactic: rw [ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto, not_forall]
state_before: case intro.intro.intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))
state_after: case intro.intro.intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∃ x_1, ¬Tendsto (fun i => ∫⁻ (ω : X), ↑(x_1 ω) ∂↑(diracProba i)) L (𝓝 (∫⁻ (ω : X), ↑(x_1 ω) ∂↑(diracProba x)))

tactic: use f
state_before: case intro.intro.intro.intro
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∃ x_1, ¬Tendsto (fun i => ∫⁻ (ω : X), ↑(x_1 ω) ∂↑(diracProba i)) L (𝓝 (∫⁻ (ω : X), ↑(x_1 ω) ∂↑(diracProba x)))
state_after: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) L (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))

tactic: simp only [diracProba, ProbabilityMeasure.coe_mk, fx_eq_one,
           lintegral_dirac' _ (measurable_coe_nnreal_ennreal_iff.mpr f.continuous.measurable)]
state_before: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto (fun i => ∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba i)) L (𝓝 (∫⁻ (ω : X), ↑(f ω) ∂↑(diracProba x)))
state_after: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto (fun i => ↑(f i)) L (𝓝 ↑1)

tactic: apply not_tendsto_iff_exists_frequently_nmem.mpr
state_before: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ¬Tendsto (fun i => ↑(f i)) L (𝓝 ↑1)
state_after: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∃ s ∈ 𝓝 ↑1, ∃ᶠ (x : X) in L, ↑(f x) ∉ s

tactic: refine ⟨Ioi 0, Ioi_mem_nhds (by simp only [ENNReal.coe_one, zero_lt_one]),
        hU.mp (eventually_of_forall ?_)⟩
state_before: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∃ s ∈ 𝓝 ↑1, ∃ᶠ (x : X) in L, ↑(f x) ∉ s
state_after: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∀ x ∉ U, ↑(f x) ∉ Ioi 0

tactic: intro x x_notin_U
state_before: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ ∀ x ∉ U, ↑(f x) ∉ Ioi 0
state_after: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhd : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
⊢ ↑(f x) ∉ Ioi 0

tactic: rw [f_vanishes_outside x
      (compl_subset_compl.mpr (show interior U ⊆ U from interior_subset) x_notin_U)]
state_before: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhd : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
⊢ ↑(f x) ∉ Ioi 0
state_after: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhd : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
⊢ ↑0 ∉ Ioi 0

tactic: simp only [ENNReal.coe_zero, mem_Ioi, lt_self_iff_false, not_false_eq_true]
state_before: case h
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x✝ : X
L : Filter X
h : ¬Tendsto id L (𝓝 x✝)
U : Set X
U_nhd : U ∈ 𝓝 x✝
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x✝
f : X →ᵇ ℝ≥0
fx_eq_one : f x✝ = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
x : X
x_notin_U : x ∉ U
⊢ ↑0 ∉ Ioi 0
state_after: no goals

tactic: by_contra! con
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
⊢ ∃ U ∈ 𝓝 x, ∃ᶠ (x : X) in L, x ∉ U
state_after: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
con : ∀ U ∈ 𝓝 x, ¬∃ᶠ (x : X) in L, x ∉ U
⊢ False

tactic: apply h
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
con : ∀ U ∈ 𝓝 x, ¬∃ᶠ (x : X) in L, x ∉ U
⊢ False
state_after: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
con : ∀ U ∈ 𝓝 x, ¬∃ᶠ (x : X) in L, x ∉ U
⊢ Tendsto id L (𝓝 x)

tactic: intro U U_nhd
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
con : ∀ U ∈ 𝓝 x, ¬∃ᶠ (x : X) in L, x ∉ U
⊢ Tendsto id L (𝓝 x)
state_after: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
con : ∀ U ∈ 𝓝 x, ¬∃ᶠ (x : X) in L, x ∉ U
U : Set X
U_nhd : U ∈ 𝓝 x
⊢ U ∈ map id L

tactic: simpa only [not_frequently, not_not] using con U U_nhd
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
con : ∀ U ∈ 𝓝 x, ¬∃ᶠ (x : X) in L, x ∉ U
U : Set X
U_nhd : U ∈ 𝓝 x
⊢ U ∈ map id L
state_after: no goals

tactic: simpa only [interior_mem_nhds] using U_nhd
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
⊢ interior U ∈ 𝓝 x
state_after: no goals

tactic: simpa only [mem_compl_iff, not_not] using mem_of_mem_nhds Uint_nhd
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
⊢ ?m.20353 ∉ (interior ?m.20123)ᶜ
state_after: no goals

tactic: simp only [ENNReal.coe_one, zero_lt_one]
state_before: X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
Uint_nhd : interior U ∈ 𝓝 x
f : X →ᵇ ℝ≥0
fx_eq_one : f x = 1
f_vanishes_outside : ∀ y ∈ (interior U)ᶜ, f y = 0
⊢ 0 < ↑1
state_after: no goals