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

tactic: by_cases hnm : m ≤ m0
state_before: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

tactic: swap
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

tactic: by_cases hfint : Integrable f μ
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

tactic: swap
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R

tactic: by_contra h
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : ¬∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
⊢ False

tactic: change μ _ ≠ 0 at h
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : ¬∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
⊢ False
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : μ {x | (fun x => |(μ[f|m]) x| ≤ ↑R) x}ᶜ ≠ 0
⊢ False

tactic: simp only [← zero_lt_iff, Set.compl_def, Set.mem_setOf_eq, not_le] at h
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : μ {x | (fun x => |(μ[f|m]) x| ≤ ↑R) x}ᶜ ≠ 0
⊢ False
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ False

tactic: suffices (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R < (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R by
  exact this.ne rfl
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ False
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R < (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R

tactic: refine lt_of_lt_of_le (setIntegral_gt_gt R.coe_nonneg ?_ ?_ h.ne.symm) ?_
state_before: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R < (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R
state_after: case pos.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ Measurable fun x => |(μ[f|m]) x|

case pos.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ IntegrableOn (fun x => |(μ[f|m]) x|) {x | ↑R < |(μ[f|m]) x|} μ

case pos.refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |(μ[f|m]) x| ∂μ ≤ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R

tactic: refine (setIntegral_abs_condexp_le ?_ _).trans ?_
state_before: case pos.refine_3
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |(μ[f|m]) x| ∂μ ≤ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R
state_after: case pos.refine_3.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ MeasurableSet {x | ↑R < |(μ[f|m]) x|}

case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |f x| ∂μ ≤ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R

tactic: simp only [← smul_eq_mul, ← setIntegral_const, NNReal.val_eq_coe, RCLike.ofReal_real_eq_id,
  _root_.id]
state_before: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |f x| ∂μ ≤ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R
state_after: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |f x| ∂μ ≤ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, ↑R ∂μ

tactic: refine setIntegral_mono_ae hfint.abs.integrableOn ?_ hbdd
state_before: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |f x| ∂μ ≤ ∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, ↑R ∂μ
state_after: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ IntegrableOn (fun x => ↑R) {x | ↑R < |(μ[f|m]) x|} μ

tactic: refine ⟨aestronglyMeasurable_const, lt_of_le_of_lt ?_
  (integrable_condexp.integrableOn : IntegrableOn (μ[f|m]) {x | ↑R < |(μ[f|m]) x|} μ).2⟩
state_before: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ IntegrableOn (fun x => ↑R) {x | ↑R < |(μ[f|m]) x|} μ
state_after: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫⁻ (a : α) in {x | ↑R < |(μ[f|m]) x|}, ↑‖(fun x => ↑R) a‖₊ ∂μ ≤
    ∫⁻ (a : α) in {x | ↑R < |(μ[f|m]) x|}, ↑‖(μ[f|m]) a‖₊ ∂μ

tactic: refine setLIntegral_mono measurable_const.nnnorm.coe_nnreal_ennreal
  (stronglyMeasurable_condexp.mono hnm).measurable.nnnorm.coe_nnreal_ennreal fun x hx => ?_
state_before: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ ∫⁻ (a : α) in {x | ↑R < |(μ[f|m]) x|}, ↑‖(fun x => ↑R) a‖₊ ∂μ ≤
    ∫⁻ (a : α) in {x | ↑R < |(μ[f|m]) x|}, ↑‖(μ[f|m]) a‖₊ ∂μ
state_after: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
x : α
hx : x ∈ {x | ↑R < |(μ[f|m]) x|}
⊢ ↑‖(fun x => ↑R) x‖₊ ≤ ↑‖(μ[f|m]) x‖₊

tactic: rw [ENNReal.coe_le_coe, Real.nnnorm_of_nonneg R.coe_nonneg]
state_before: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
x : α
hx : x ∈ {x | ↑R < |(μ[f|m]) x|}
⊢ ↑‖(fun x => ↑R) x‖₊ ≤ ↑‖(μ[f|m]) x‖₊
state_after: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
x : α
hx : x ∈ {x | ↑R < |(μ[f|m]) x|}
⊢ ⟨↑R, ⋯⟩ ≤ ‖(μ[f|m]) x‖₊

tactic: exact Subtype.mk_le_mk.2 (le_of_lt hx)
state_before: case pos.refine_3.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
x : α
hx : x ∈ {x | ↑R < |(μ[f|m]) x|}
⊢ ⟨↑R, ⋯⟩ ≤ ‖(μ[f|m]) x‖₊
state_after: no goals

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

tactic: exact eventually_of_forall fun _ => R.coe_nonneg
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : ¬m ≤ m0
⊢ ∀ᵐ (x : α) ∂μ, 0 ≤ ↑R
state_after: no goals

tactic: simp_rw [condexp_undef hfint]
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |(μ[f|m]) x| ≤ ↑R
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |0 x| ≤ ↑R

tactic: filter_upwards [hbdd] with x hx
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∀ᵐ (x : α) ∂μ, |0 x| ≤ ↑R
state_after: case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
x : α
hx : |f x| ≤ ↑R
⊢ |0 x| ≤ ↑R

tactic: rw [Pi.zero_apply, abs_zero]
state_before: case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
x : α
hx : |f x| ≤ ↑R
⊢ |0 x| ≤ ↑R
state_after: case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
x : α
hx : |f x| ≤ ↑R
⊢ 0 ≤ ↑R

tactic: exact (abs_nonneg _).trans hx
state_before: case h
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : ¬Integrable f μ
x : α
hx : |f x| ≤ ↑R
⊢ 0 ≤ ↑R
state_after: no goals

tactic: exact this.ne rfl
state_before: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
this : (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R < (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R
⊢ False
state_after: no goals

tactic: simp_rw [← Real.norm_eq_abs]
state_before: case pos.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ Measurable fun x => |(μ[f|m]) x|
state_after: case pos.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ Measurable fun x => ‖(μ[f|m]) x‖

tactic: exact (stronglyMeasurable_condexp.mono hnm).measurable.norm
state_before: case pos.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ Measurable fun x => ‖(μ[f|m]) x‖
state_after: no goals

tactic: exact integrable_condexp.abs.integrableOn
state_before: case pos.refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ IntegrableOn (fun x => |(μ[f|m]) x|) {x | ↑R < |(μ[f|m]) x|} μ
state_after: no goals

tactic: simp_rw [← Real.norm_eq_abs]
state_before: case pos.refine_3.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ MeasurableSet {x | ↑R < |(μ[f|m]) x|}
state_after: case pos.refine_3.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ MeasurableSet {x | ↑R < ‖(μ[f|m]) x‖}

tactic: exact @measurableSet_lt _ _ _ _ _ m _ _ _ _ _ measurable_const
  stronglyMeasurable_condexp.norm.measurable
state_before: case pos.refine_3.refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
R : ℝ≥0
f : α → ℝ
hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R
hnm : m ≤ m0
hfint : Integrable f μ
h : 0 < μ {x | ↑R < |(μ[f|m]) x|}
⊢ MeasurableSet {x | ↑R < ‖(μ[f|m]) x‖}
state_after: no goals