Theorem: IsFiniteMeasureOnCompacts.map
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

tactic: intro K hK
state_before: α✝ : Type ?u.1653244
β✝ : Type ?u.1653247
γ : Type ?u.1653250
γ₂ : Type ?u.1653253
δ : Type ?u.1653256
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_1
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_2
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : MeasureTheory.Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
⊢ ∀ ⦃K : Set β⦄, IsCompact K → ↑↑(Measure.map (↑f) μ) K < ⊤

tactic: rw [Measure.map_apply f.measurable hK.measurableSet]
state_before: α✝ : Type ?u.1653244
β✝ : Type ?u.1653247
γ : Type ?u.1653250
γ₂ : Type ?u.1653253
δ : Type ?u.1653256
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_1
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_2
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : MeasureTheory.Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ ↑↑(Measure.map (↑f) μ) K < ⊤

tactic: apply IsCompact.measure_lt_top
state_before: α✝ : Type ?u.1653244
β✝ : Type ?u.1653247
γ : Type ?u.1653250
γ₂ : Type ?u.1653253
δ : Type ?u.1653256
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_1
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_2
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : MeasureTheory.Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ ↑↑μ (↑f ⁻¹' K) < ⊤

tactic: rwa [f.isCompact_preimage]
state_before: case hK
α✝ : Type ?u.1653244
β✝ : Type ?u.1653247
γ : Type ?u.1653250
γ₂ : Type ?u.1653253
δ : Type ?u.1653256
ι : Sort y
s t u : Set α✝
inst✝¹⁶ : TopologicalSpace α✝
inst✝¹⁵ : MeasurableSpace α✝
inst✝¹⁴ : BorelSpace α✝
inst✝¹³ : TopologicalSpace β✝
inst✝¹² : MeasurableSpace β✝
inst✝¹¹ : BorelSpace β✝
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : MeasurableSpace γ
inst✝⁸ : BorelSpace γ
inst✝⁷ : MeasurableSpace δ
α : Type u_1
m0 : MeasurableSpace α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OpensMeasurableSpace α
β : Type u_2
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : BorelSpace β
inst✝¹ : T2Space β
μ : MeasureTheory.Measure α
inst✝ : IsFiniteMeasureOnCompacts μ
f : α ≃ₜ β
K : Set β
hK : IsCompact K
⊢ IsCompact (↑f ⁻¹' K)