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

Proof steps:
state_before: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
tactic: let fs := hf.approxBounded c
state_before: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : μ = 0
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]

case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
tactic: by_cases hμ : μ = 0
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this : (ae μ).NeBot
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
tactic: have : (ae μ).NeBot := ae_neBot.2 hμ
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this : (ae μ).NeBot
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this : (ae μ).NeBot
hc : 0 ≤ c
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
tactic: have hc : 0 ≤ c := by
    rcases hf_bound.exists with ⟨_x, hx⟩
    exact (norm_nonneg _).trans hx
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this : (ae μ).NeBot
hc : 0 ≤ c
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (x : α), ‖↑(fs n) x‖ ≤ c
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
tactic: have hfs_bound : ∀ n x, ‖fs n x‖ ≤ c := hf.norm_approxBounded_le hc
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (x : α), ‖↑(fs n) x‖ ≤ c
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (x : α), ‖↑(fs n) x‖ ≤ c
this : μ[f * μ[g|m]|m] = f * μ[g|m]
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
tactic: have : μ[f * μ[g|m]|m] = f * μ[g|m] := by
    refine condexp_of_stronglyMeasurable hm (hf.mul stronglyMeasurable_condexp) ?_
    exact integrable_condexp.bdd_mul' (hf.mono hm).aestronglyMeasurable hf_bound
state_before: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (x : α), ‖↑(fs n) x‖ ≤ c
this : μ[f * μ[g|m]|m] = f * μ[g|m]
⊢ μ[f * g|m] =ᶠ[ae μ] f * μ[g|m]
state_after: case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : IsFiniteMeasure μ
f g : α → ℝ
hf : StronglyMeasurable f
hg : Integrable g μ
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
fs : ℕ → SimpleFunc α ℝ := hf.approxBounded c
hfs_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 => ↑(fs x_1) x) atTop (𝓝 (f x))
hμ : ¬μ = 0
this✝ : (ae μ).NeBot
hc : 0 ≤ c
hfs_bound : ∀ (n : ℕ) (x : α), ‖↑(fs n) x‖ ≤ c
this : μ[f * μ[g|m]|m] = f * μ[g|m]
⊢ μ[f * g|m] =ᶠ[ae μ] μ[f * μ[g|m]|m]
tactic: rw [← this]