File: Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean

full_name: BoundedContinuousFunction.apply_le_nndist_zero
code: lemma apply_le_nndist_zero {X : Type*} [TopologicalSpace X] (f : X →ᵇ ℝ≥0) (x : X) :
    f x ≤ nndist 0 f 
goal: f x ≤ nndist 0 f

full_name: BoundedContinuousFunction.lintegral_le_edist_mul
code: lemma lintegral_le_edist_mul (f : X →ᵇ ℝ≥0) (μ : Measure X) :
    (∫⁻ x, f x ∂μ) ≤ edist 0 f * (μ Set.univ) 
goal: ∫⁻ (x : X), ↑(f x) ∂μ ≤ edist 0 f * μ Set.univ

full_name: BoundedContinuousFunction.measurable_coe_ennreal_comp
code: theorem measurable_coe_ennreal_comp (f : X →ᵇ ℝ≥0) :
    Measurable fun x ↦ (f x : ℝ≥0∞) 
goal: Measurable fun x => ↑(f x)

full_name: BoundedContinuousFunction.lintegral_lt_top_of_nnreal
code: theorem lintegral_lt_top_of_nnreal (f : X →ᵇ ℝ≥0) : ∫⁻ x, f x ∂μ < ∞ 
goal: ∫⁻ (x : X), ↑(f x) ∂μ < ⊤

full_name: BoundedContinuousFunction.integrable_of_nnreal
code: theorem integrable_of_nnreal (f : X →ᵇ ℝ≥0) : Integrable (((↑) : ℝ≥0 → ℝ) ∘ ⇑f) μ 
goal: MeasureTheory.Integrable (NNReal.toReal ∘ ⇑f) μ

full_name: BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub
code: theorem integral_eq_integral_nnrealPart_sub (f : X →ᵇ ℝ) :
    ∫ x, f x ∂μ = (∫ x, (f.nnrealPart x : ℝ) ∂μ) - ∫ x, ((-f).nnrealPart x : ℝ) ∂μ 
goal: ∫ (x : X), f x ∂μ = ∫ (x : X), ↑(f.nnrealPart x) ∂μ - ∫ (x : X), ↑((-f).nnrealPart x) ∂μ

full_name: BoundedContinuousFunction.lintegral_of_real_lt_top
code: theorem lintegral_of_real_lt_top (f : X →ᵇ ℝ) :
    ∫⁻ x, ENNReal.ofReal (f x) ∂μ < ∞ 
goal: ∫⁻ (x : X), ENNReal.ofReal (f x) ∂μ < ⊤

full_name: BoundedContinuousFunction.toReal_lintegral_coe_eq_integral
code: theorem toReal_lintegral_coe_eq_integral (f : X →ᵇ ℝ≥0) (μ : Measure X) :
    (∫⁻ x, (f x : ℝ≥0∞) ∂μ).toReal = ∫ x, (f x : ℝ) ∂μ 
goal: (∫⁻ (x : X), ↑(f x) ∂μ).toReal = ∫ (x : X), ↑(f x) ∂μ

full_name: BoundedContinuousFunction.lintegral_nnnorm_le
code: lemma lintegral_nnnorm_le (f : X →ᵇ E) :
    ∫⁻ x, ‖f x‖₊ ∂μ ≤ ‖f‖₊ * (μ Set.univ) 
goal: ∫⁻ (x : X), ↑‖f x‖₊ ∂μ ≤ ↑‖f‖₊ * μ Set.univ

full_name: BoundedContinuousFunction.integrable
code: lemma integrable [IsFiniteMeasure μ] (f : X →ᵇ E) :
    Integrable f μ 
goal: MeasureTheory.Integrable (⇑f) μ

full_name: BoundedContinuousFunction.norm_integral_le_mul_norm
code: lemma norm_integral_le_mul_norm [IsFiniteMeasure μ] (f : X →ᵇ E) :
    ‖∫ x, f x ∂μ‖ ≤ ENNReal.toReal (μ Set.univ) * ‖f‖ 
goal: ‖∫ (x : X), f x ∂μ‖ ≤ (μ Set.univ).toReal * ‖f‖

full_name: BoundedContinuousFunction.norm_integral_le_norm
code: lemma norm_integral_le_norm [IsProbabilityMeasure μ] (f : X →ᵇ E) :
    ‖∫ x, f x ∂μ‖ ≤ ‖f‖ 
goal: ‖∫ (x : X), f x ∂μ‖ ≤ ‖f‖

full_name: BoundedContinuousFunction.isBounded_range_integral
code: lemma isBounded_range_integral
    {ι : Type*} (μs : ι → Measure X) [∀ i, IsProbabilityMeasure (μs i)] (f : X →ᵇ E) :
    Bornology.IsBounded (Set.range (fun i ↦ ∫ x, f x ∂ (μs i))) 
goal: Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)

full_name: BoundedContinuousFunction.integral_add_const
code: lemma integral_add_const (f : X →ᵇ ℝ) (c : ℝ) :
    ∫ x, (f + const X c) x ∂μ = ∫ x, f x ∂μ + ENNReal.toReal (μ (Set.univ)) • c 
goal: ∫ (x : X), (f + BoundedContinuousFunction.const X c) x ∂μ = ∫ (x : X), f x ∂μ + (μ Set.univ).toReal • c

full_name: BoundedContinuousFunction.integral_const_sub
code: lemma integral_const_sub (f : X →ᵇ ℝ) (c : ℝ) :
    ∫ x, (const X c - f) x ∂μ = ENNReal.toReal (μ (Set.univ)) • c - ∫ x, f x ∂μ 
goal: ∫ (x : X), (BoundedContinuousFunction.const X c - f) x ∂μ = (μ Set.univ).toReal • c - ∫ (x : X), f x ∂μ

full_name: BoundedContinuousFunction.tendsto_integral_of_forall_limsup_integral_le_integral
code: lemma tendsto_integral_of_forall_limsup_integral_le_integral {ι : Type*} {L : Filter ι}
    {μ : Measure X} [IsProbabilityMeasure μ] {μs : ι → Measure X} [∀ i, IsProbabilityMeasure (μs i)]
    (h : ∀ f : X →ᵇ ℝ, 0 ≤ f → L.limsup (fun i ↦ ∫ x, f x ∂ (μs i)) ≤ ∫ x, f x ∂μ)
    (f : X →ᵇ ℝ) :
    Tendsto (fun i ↦ ∫ x, f x ∂ (μs i)) L (𝓝 (∫ x, f x ∂μ)) 
goal: Filter.Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))

full_name: BoundedContinuousFunction.tendsto_integral_of_forall_integral_le_liminf_integral
code: lemma tendsto_integral_of_forall_integral_le_liminf_integral {ι : Type*} {L : Filter ι}
    {μ : Measure X} [IsProbabilityMeasure μ] {μs : ι → Measure X} [∀ i, IsProbabilityMeasure (μs i)]
    (h : ∀ f : X →ᵇ ℝ, 0 ≤ f → ∫ x, f x ∂μ ≤ L.liminf (fun i ↦ ∫ x, f x ∂ (μs i)))
    (f : X →ᵇ ℝ) :
    Tendsto (fun i ↦ ∫ x, f x ∂ (μs i)) L (𝓝 (∫ x, f x ∂μ)) 
goal: Filter.Tendsto (fun i => ∫ (x : X), f x ∂μs i) L (𝓝 (∫ (x : X), f x ∂μ))