l3lab/ntp-mathlib huggingface dataset:
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by 

Proof state:
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
⊢ ∀ (x : ℝ≥0∞), (fun x x_1 => x ≥ x_1) x 0

Next tactic:
simp only [ge_iff_le, zero_le, forall_const]
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · 

Proof state:
case h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
⊢ ∀ (b : ℝ≥0∞), (∀ᶠ (n : ℕ) in atTop, b ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n) → b ≤ ↑‖f‖₊

Next tactic:
intro x hx
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      

Proof state:
case h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
⊢ x ≤ ↑‖f‖₊

Next tactic:
obtain ⟨i, hi⟩ := hx.exists
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      obtain ⟨i, hi⟩ := hx.exists
      

Proof state:
case h.intro
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
⊢ x ≤ ↑‖f‖₊

Next tactic:
apply le_trans hi
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      obtain ⟨i, hi⟩ := hx.exists
      apply le_trans hi
      

Proof state:
case h.intro
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
⊢ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i ≤ ↑‖f‖₊

Next tactic:
convert obs i with x
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      obtain ⟨i, hi⟩ := hx.exists
      apply le_trans hi
      convert obs i with x
      

Proof state:
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x✝ : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
x : Ω
⊢ ENNReal.ofReal (f x) = ↑‖f x‖₊

Next tactic:
have aux := ENNReal.ofReal_eq_coe_nnreal (f_nn x)
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      obtain ⟨i, hi⟩ := hx.exists
      apply le_trans hi
      convert obs i with x
      have aux := ENNReal.ofReal_eq_coe_nnreal (f_nn x)
      

Proof state:
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x✝ : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
x : Ω
aux :
  ENNReal.ofReal ((fun f => f.toFun) f x) =
    ↑{ val := (fun f => f.toFun) f x, property := (_ : (fun f => f.toFun) 0 x ≤ (fun f => f.toFun) f x) }
⊢ ENNReal.ofReal (f x) = ↑‖f x‖₊

Next tactic:
simp only [ContinuousMap.toFun_eq_coe, BoundedContinuousFunction.coe_to_continuous_fun] at aux
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      obtain ⟨i, hi⟩ := hx.exists
      apply le_trans hi
      convert obs i with x
      have aux := ENNReal.ofReal_eq_coe_nnreal (f_nn x)
      simp only [ContinuousMap.toFun_eq_coe, BoundedContinuousFunction.coe_to_continuous_fun] at aux
      

Proof state:
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x✝ : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
x : Ω
aux : ENNReal.ofReal (f x) = ↑{ val := f x, property := (_ : (fun r => 0 ≤ r) (f x)) }
⊢ ENNReal.ofReal (f x) = ↑‖f x‖₊

Next tactic:
rw [aux]
lemma integral_le_liminf_integral_of_forall_isOpen_measure_le_liminf_measure
    {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ℕ → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
    {f : Ω →ᵇ ℝ} (f_nn : 0 ≤ f)
    (h_opens : ∀ G, IsOpen G → μ G ≤ atTop.liminf (fun i ↦ μs i G)) :
    ∫ x, (f x) ∂μ ≤ atTop.liminf (fun i ↦ ∫ x, (f x) ∂ (μs i)) := by
  have same := lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
                  f.continuous f_nn h_opens
  rw [@integral_eq_lintegral_of_nonneg_ae Ω _ μ f (eventually_of_forall f_nn)
        f.continuous.measurable.aestronglyMeasurable]
  convert (ENNReal.toReal_le_toReal ?_ ?_).mpr same
  · simp only [fun i ↦ @integral_eq_lintegral_of_nonneg_ae Ω _ (μs i) f (eventually_of_forall f_nn)
                        f.continuous.measurable.aestronglyMeasurable]
    let g := BoundedContinuousFunction.comp _ Real.lipschitzWith_toNNReal f
    have bound : ∀ i, ∫⁻ x, ENNReal.ofReal (f x) ∂(μs i) ≤ nndist 0 g := fun i ↦ by
      simpa only [coe_nnreal_ennreal_nndist, measure_univ, mul_one, ge_iff_le] using
            BoundedContinuousFunction.lintegral_le_edist_mul (μ := μs i) g
    apply ENNReal.liminf_toReal_eq ENNReal.coe_ne_top (eventually_of_forall bound)
  · exact (f.lintegral_of_real_lt_top μ).ne
  · apply ne_of_lt
    have obs := fun (i : ℕ) ↦ @BoundedContinuousFunction.lintegral_nnnorm_le Ω _ _ (μs i) ℝ _ f
    simp only [measure_univ, mul_one] at obs
    apply lt_of_le_of_lt _ (show (‖f‖₊ : ℝ≥0∞) < ∞ from ENNReal.coe_lt_top)
    apply liminf_le_of_le
    · refine ⟨0, eventually_of_forall (by simp only [ge_iff_le, zero_le, forall_const])⟩
    · intro x hx
      obtain ⟨i, hi⟩ := hx.exists
      apply le_trans hi
      convert obs i with x
      have aux := ENNReal.ofReal_eq_coe_nnreal (f_nn x)
      simp only [ContinuousMap.toFun_eq_coe, BoundedContinuousFunction.coe_to_continuous_fun] at aux
      rw [aux]
      

Proof state:
case h.e'_3.h.e'_4.h
Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsProbabilityMeasure μ
μs : ℕ → Measure Ω
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)
f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
h_opens : ∀ (G : Set Ω), IsOpen G → ↑↑μ G ≤ liminf (fun i => ↑↑(μs i) G) atTop
same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop
obs : ∀ (i : ℕ), ∫⁻ (x : Ω), ↑‖f x‖₊ ∂μs i ≤ ↑‖f‖₊
x✝ : ℝ≥0∞
hx : ∀ᶠ (n : ℕ) in atTop, x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs n
i : ℕ
hi : x✝ ≤ ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i
x : Ω
aux : ENNReal.ofReal (f x) = ↑{ val := f x, property := (_ : (fun r => 0 ≤ r) (f x)) }
⊢ ↑{ val := f x, property := (_ : (fun r => 0 ≤ r) (f x)) } = ↑‖f x‖₊

Next tactic:
congr