-- Mathlib/MeasureTheory/Measure/DiracProba.lean
lemma not_tendsto_diracProba_of_not_tendsto [CompletelyRegularSpace X] {x : X} (L : Filter X)
    (h : ¬ Tendsto id L (𝓝 x)) :
    ¬ Tendsto diracProba L (𝓝 (diracProba x))

Proof state:
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 ?_)⟩

Result:
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

Proof state:
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]

Result:
no goals

Proof state:
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

Result:
no goals

Proof state:
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)

Result:
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))

Proof state:
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)]

Result:
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)

Proof state:
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]

Result:
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)))

Proof state:
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

Tactic:
simp only [ENNReal.coe_one, zero_lt_one]

Result:
no goals

Proof state:
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

Tactic:
simpa only [interior_mem_nhds] using U_nhd

Result:
no goals