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

Proof steps:
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: by_cases hm : m ≤ m0
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: 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
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: 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
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: 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
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: rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]