feat(measure_theory/measure/probability_measure): Add a bit of simple…

… API to probability_measure. (#17610)

Add a bit of simple API to `probability_measure` and `finite_measure`.

src/measure_theory/measure/finite_measure.lean

@@ -119,6 +119,14 @@ lemma coe_fn_eq_to_nnreal_coe_fn_to_measure (ν : finite_measure Ω) :
 lemma coe_injective : function.injective (coe : finite_measure Ω → measure Ω) :=
 subtype.coe_injective
 
+lemma apply_mono (μ : finite_measure Ω) {s₁ s₂ : set Ω} (h : s₁ ⊆ s₂) :
+  μ s₁ ≤ μ s₂ :=
+begin
+  change ((μ : measure Ω) s₁).to_nnreal ≤ ((μ : measure Ω) s₂).to_nnreal,
+  have key : (μ : measure Ω) s₁ ≤ (μ : measure Ω) s₂ := (μ : measure Ω).mono h,
+  apply (ennreal.to_nnreal_le_to_nnreal (measure_ne_top _ s₁) (measure_ne_top _ s₂)).mpr key,
+end
+
 /-- The (total) mass of a finite measure `μ` is `μ univ`, i.e., the cast to `nnreal` of
 `(μ : measure Ω) univ`. -/
 def mass (μ : finite_measure Ω) : ℝ≥0 := μ univ
@@ -145,11 +153,16 @@ begin
   exact finite_measure.mass_zero_iff μ,
 end
 
-@[ext] lemma extensionality (μ ν : finite_measure Ω)
+@[ext] lemma eq_of_forall_measure_apply_eq (μ ν : finite_measure Ω)
+  (h : ∀ (s : set Ω), measurable_set s → (μ : measure Ω) s = (ν : measure Ω) s) :
+  μ = ν :=
+by { ext1, ext1 s s_mble, exact h s s_mble, }
+
+lemma eq_of_forall_apply_eq (μ ν : finite_measure Ω)
   (h : ∀ (s : set Ω), measurable_set s → μ s = ν s) :
   μ = ν :=
 begin
-  ext1, ext1 s s_mble,
+  ext1 s s_mble,
   simpa [ennreal_coe_fn_eq_coe_fn_to_measure] using congr_arg (coe : ℝ≥0 → ℝ≥0∞) (h s s_mble),
 end
 
@@ -198,6 +211,39 @@ function.injective.module _ coe_add_monoid_hom coe_injective coe_smul
   (c • μ) s  = c • (μ s) :=
 by { simp only [coe_fn_smul, pi.smul_apply], }
 
+/-- Restrict a finite measure μ to a set A. -/
+def restrict (μ : finite_measure Ω) (A : set Ω) : finite_measure Ω :=
+{ val := (μ : measure Ω).restrict A,
+  property := measure_theory.is_finite_measure_restrict μ A, }
+
+lemma restrict_measure_eq (μ : finite_measure Ω) (A : set Ω) :
+  (μ.restrict A : measure Ω) = (μ : measure Ω).restrict A := rfl
+
+lemma restrict_apply_measure (μ : finite_measure Ω) (A : set Ω)
+  {s : set Ω} (s_mble : measurable_set s) :
+  (μ.restrict A : measure Ω) s = (μ : measure Ω) (s ∩ A) :=
+measure.restrict_apply s_mble
+
+lemma restrict_apply (μ : finite_measure Ω) (A : set Ω)
+  {s : set Ω} (s_mble : measurable_set s) :
+  (μ.restrict A) s = μ (s ∩ A) :=
+begin
+  apply congr_arg ennreal.to_nnreal,
+  exact measure.restrict_apply s_mble,
+end
+
+lemma restrict_mass (μ : finite_measure Ω) (A : set Ω) :
+  (μ.restrict A).mass = μ A :=
+by simp only [mass, restrict_apply μ A measurable_set.univ, univ_inter]
+
+lemma restrict_eq_zero_iff (μ : finite_measure Ω) (A : set Ω) :
+  μ.restrict A = 0 ↔ μ A = 0 :=
+by rw [← mass_zero_iff, restrict_mass]
+
+lemma restrict_nonzero_iff (μ : finite_measure Ω) (A : set Ω) :
+  μ.restrict A ≠ 0 ↔ μ A ≠ 0 :=
+by rw [← mass_nonzero_iff, restrict_mass]
+
 variables [topological_space Ω]
 
 /-- The pairing of a finite (Borel) measure `μ` with a nonnegative bounded continuous

src/measure_theory/measure/probability_measure.lean

@@ -5,6 +5,7 @@ Authors: Kalle Kytölä
 -/
 import measure_theory.measure.finite_measure
 import measure_theory.integral.average
+import probability.conditional_probability
 
 /-!
 # Probability measures
@@ -116,6 +117,9 @@ subtype.coe_injective
 @[simp] lemma coe_fn_univ (ν : probability_measure Ω) : ν univ = 1 :=
 congr_arg ennreal.to_nnreal ν.prop.measure_univ
 
+lemma coe_fn_univ_ne_zero (ν : probability_measure Ω) : ν univ ≠ 0 :=
+by simp only [coe_fn_univ, ne.def, one_ne_zero, not_false_iff]
+
 /-- A probability measure can be interpreted as a finite measure. -/
 def to_finite_measure (μ : probability_measure Ω) : finite_measure Ω := ⟨μ, infer_instance⟩
 
@@ -130,11 +134,32 @@ def to_finite_measure (μ : probability_measure Ω) : finite_measure Ω := ⟨μ
 by rw [← coe_fn_comp_to_finite_measure_eq_coe_fn,
   finite_measure.ennreal_coe_fn_eq_coe_fn_to_measure, coe_comp_to_finite_measure_eq_coe]
 
-@[ext] lemma extensionality (μ ν : probability_measure Ω)
+lemma apply_mono (μ : probability_measure Ω) {s₁ s₂ : set Ω} (h : s₁ ⊆ s₂) :
+  μ s₁ ≤ μ s₂ :=
+begin
+  rw ← coe_fn_comp_to_finite_measure_eq_coe_fn,
+  exact measure_theory.finite_measure.apply_mono _ h,
+end
+
+lemma nonempty_of_probability_measure (μ : probability_measure Ω) : nonempty Ω :=
+begin
+  by_contra maybe_empty,
+  have zero : (μ : measure Ω) univ = 0,
+    by rw [univ_eq_empty_iff.mpr (not_nonempty_iff.mp maybe_empty), measure_empty],
+  rw measure_univ at zero,
+  exact zero_ne_one zero.symm,
+end
+
+@[ext] lemma eq_of_forall_measure_apply_eq (μ ν : probability_measure Ω)
+  (h : ∀ (s : set Ω), measurable_set s → (μ : measure Ω) s = (ν : measure Ω) s) :
+  μ = ν :=
+by { ext1, ext1 s s_mble, exact h s s_mble, }
+
+lemma eq_of_forall_apply_eq (μ ν : probability_measure Ω)
   (h : ∀ (s : set Ω), measurable_set s → μ s = ν s) :
   μ = ν :=
 begin
-  ext1, ext1 s s_mble,
+  ext1 s s_mble,
   simpa [ennreal_coe_fn_eq_coe_fn_to_measure] using congr_arg (coe : ℝ≥0 → ℝ≥0∞) (h s s_mble),
 end
 
@@ -268,7 +293,8 @@ end
 
 lemma self_eq_mass_smul_normalize : μ = μ.mass • μ.normalize.to_finite_measure :=
 begin
-  ext s s_mble,
+  apply eq_of_forall_apply_eq,
+  intros s s_mble,
   rw [μ.self_eq_mass_mul_normalize s, coe_fn_smul_apply, smul_eq_mul,
     probability_measure.coe_fn_comp_to_finite_measure_eq_coe_fn],
 end
@@ -299,10 +325,11 @@ end
   {m0 : measurable_space Ω} (μ : probability_measure Ω) :
   μ.to_finite_measure.normalize = μ :=
 begin
-  ext s s_mble,
+  apply probability_measure.eq_of_forall_apply_eq,
+  intros s s_mble,
   rw μ.to_finite_measure.normalize_eq_of_nonzero μ.to_finite_measure_nonzero s,
   simp only [probability_measure.mass_to_finite_measure, inv_one, one_mul,
-    probability_measure.coe_fn_comp_to_finite_measure_eq_coe_fn],
+             probability_measure.coe_fn_comp_to_finite_measure_eq_coe_fn],
 end
 
 /-- Averaging with respect to a finite measure is the same as integraing against