Theorem: MeasureTheory.integral_abs_condexp_le
File: Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean

tactic: by_cases hm : m ≤ m0
state_before: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : ¬m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

tactic: swap
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : ¬m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : ¬m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

tactic: by_cases hfint : Integrable f μ
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

tactic: swap
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ

tactic: rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ (∫⁻ (a : α), ENNReal.ofReal |(μ[f|m]) a| ∂μ).toReal ≤ (∫⁻ (a : α), ENNReal.ofReal |f a| ∂μ).toReal

case pos.hf
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ 0 ≤ᶠ[ae μ] fun x => |f x|

case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => |f x|) μ

case pos.hf
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ 0 ≤ᶠ[ae μ] fun x => |(μ[f|m]) x|

case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => |(μ[f|m]) x|) μ

tactic: simp_rw [condexp_of_not_le hm, Pi.zero_apply, abs_zero, integral_zero]
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : ¬m ≤ m0
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : ¬m ≤ m0
⊢ 0 ≤ ∫ (x : α), |f x| ∂μ

tactic: positivity
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : ¬m ≤ m0
⊢ 0 ≤ ∫ (x : α), |f x| ∂μ
state_after: no goals

tactic: simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,
  mul_zero]
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α), |(μ[f|m]) x| ∂μ ≤ ∫ (x : α), |f x| ∂μ
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ 0 ≤ ∫ (x : α), |f x| ∂μ

tactic: positivity
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : ¬Integrable f μ
⊢ 0 ≤ ∫ (x : α), |f x| ∂μ
state_after: no goals

tactic: rw [ENNReal.toReal_le_toReal] <;> simp_rw [← Real.norm_eq_abs, ofReal_norm_eq_coe_nnnorm]
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ (∫⁻ (a : α), ENNReal.ofReal |(μ[f|m]) a| ∂μ).toReal ≤ (∫⁻ (a : α), ENNReal.ofReal |f a| ∂μ).toReal
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫⁻ (a : α), ↑‖(μ[f|m]) a‖₊ ∂μ ≤ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ

case pos.ha
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫⁻ (a : α), ↑‖(μ[f|m]) a‖₊ ∂μ ≠ ⊤

case pos.hb
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ ≠ ⊤

tactic: rw [← snorm_one_eq_lintegral_nnnorm, ← snorm_one_eq_lintegral_nnnorm]
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫⁻ (a : α), ↑‖(μ[f|m]) a‖₊ ∂μ ≤ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ

tactic: exact snorm_one_condexp_le_snorm _
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ snorm (μ[f|m]) 1 μ ≤ snorm f 1 μ
state_after: no goals

tactic: exact integrable_condexp.2.ne
state_before: case pos.ha
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫⁻ (a : α), ↑‖(μ[f|m]) a‖₊ ∂μ ≠ ⊤
state_after: no goals

tactic: exact hfint.2.ne
state_before: case pos.hb
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ ≠ ⊤
state_after: no goals

tactic: filter_upwards with x using abs_nonneg _
state_before: case pos.hf
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ 0 ≤ᶠ[ae μ] fun x => |f x|
state_after: no goals

tactic: simp_rw [← Real.norm_eq_abs]
state_before: case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => |f x|) μ
state_after: case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => ‖f x‖) μ

tactic: exact hfint.1.norm
state_before: case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => ‖f x‖) μ
state_after: no goals

tactic: filter_upwards with x using abs_nonneg _
state_before: case pos.hf
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ 0 ≤ᶠ[ae μ] fun x => |(μ[f|m]) x|
state_after: no goals

tactic: simp_rw [← Real.norm_eq_abs]
state_before: case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => |(μ[f|m]) x|) μ
state_after: case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => ‖(μ[f|m]) x‖) μ

tactic: exact (stronglyMeasurable_condexp.mono hm).aestronglyMeasurable.norm
state_before: case pos.hfm
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hm : m ≤ m0
hfint : Integrable f μ
⊢ AEStronglyMeasurable (fun x => ‖(μ[f|m]) x‖) μ
state_after: no goals