diff --git a/Mathlib/Algebra/Order/BigOperators/Expect.lean b/Mathlib/Algebra/Order/BigOperators/Expect.lean
index 30064a4bcb627..a449b9ce36435 100644
--- a/Mathlib/Algebra/Order/BigOperators/Expect.lean
+++ b/Mathlib/Algebra/Order/BigOperators/Expect.lean
@@ -130,8 +130,7 @@ end LinearOrderedAddCommMonoid
 section LinearOrderedAddCommGroup
 variable [LinearOrderedAddCommGroup α] [Module ℚ≥0 α] [PosSMulMono ℚ≥0 α]
 
--- TODO: Norm version
-lemma abs_expect_le_expect_abs (s : Finset ι) (f : ι → α) : |𝔼 i ∈ s, f i| ≤ 𝔼 i ∈ s, |f i| :=
+lemma abs_expect_le (s : Finset ι) (f : ι → α) : |𝔼 i ∈ s, f i| ≤ 𝔼 i ∈ s, |f i| :=
   le_expect_of_subadditive abs_zero abs_add (fun _ ↦ abs_nnqsmul _)
 
 end LinearOrderedAddCommGroup
diff --git a/Mathlib/Analysis/Normed/Group/Basic.lean b/Mathlib/Analysis/Normed/Group/Basic.lean
index bde983756cf20..1eba1978859f4 100644
--- a/Mathlib/Analysis/Normed/Group/Basic.lean
+++ b/Mathlib/Analysis/Normed/Group/Basic.lean
@@ -1129,7 +1129,7 @@ theorem nnnorm_prod_le (s : Finset ι) (f : ι → E) : ‖∏ a ∈ s, f a‖
 @[to_additive]
 theorem nnnorm_prod_le_of_le (s : Finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b) :
     ‖∏ b ∈ s, f b‖₊ ≤ ∑ b ∈ s, n b :=
-  (norm_prod_le_of_le s h).trans_eq NNReal.coe_sum.symm
+  (norm_prod_le_of_le s h).trans_eq (NNReal.coe_sum ..).symm
 
 namespace Real
 
diff --git a/Mathlib/Analysis/Normed/Group/Constructions.lean b/Mathlib/Analysis/Normed/Group/Constructions.lean
index a43e9e963c9ee..c9ee9cc5d178f 100644
--- a/Mathlib/Analysis/Normed/Group/Constructions.lean
+++ b/Mathlib/Analysis/Normed/Group/Constructions.lean
@@ -362,7 +362,7 @@ lemma Pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ Fintype.card ι • ‖f
 @[to_additive Pi.sum_nnnorm_apply_le_nnnorm "The $L^1$ norm is less than the $L^\\infty$ norm
 scaled by the cardinality."]
 lemma Pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊ :=
-  NNReal.coe_sum.trans_le <| Pi.sum_norm_apply_le_norm' _
+  (NNReal.coe_sum ..).trans_le <| Pi.sum_norm_apply_le_norm' _
 
 end SeminormedGroup
 
diff --git a/Mathlib/Analysis/RCLike/Basic.lean b/Mathlib/Analysis/RCLike/Basic.lean
index 869aa9fe0a463..82077c49fead5 100644
--- a/Mathlib/Analysis/RCLike/Basic.lean
+++ b/Mathlib/Analysis/RCLike/Basic.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Frédéric Dupuis
 -/
 import Mathlib.Algebra.Algebra.Field
+import Mathlib.Algebra.Order.BigOperators.Expect
 import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Analysis.CStarAlgebra.Basic
 import Mathlib.Analysis.Normed.Operator.ContinuousLinearMap
@@ -39,7 +40,7 @@ their counterparts in `Mathlib/Analysis/Complex/Basic.lean` (which causes linter
 A few lemmas requiring heavier imports are in `Mathlib/Data/RCLike/Lemmas.lean`.
 -/
 
-open scoped ComplexConjugate
+open scoped BigOperators ComplexConjugate
 
 section
 
@@ -234,6 +235,10 @@ theorem norm_ofReal (r : ℝ) : ‖(r : K)‖ = |r| :=
 instance (priority := 100) charZero_rclike : CharZero K :=
   (RingHom.charZero_iff (algebraMap ℝ K).injective).1 inferInstance
 
+@[rclike_simps, norm_cast]
+lemma ofReal_expect {α : Type*} (s : Finset α) (f : α → ℝ) : 𝔼 i ∈ s, f i = 𝔼 i ∈ s, (f i : K) :=
+  map_expect (algebraMap ..) ..
+
 /-! ### The imaginary unit, `I` -/
 
 /-- The imaginary unit. -/
@@ -605,13 +610,23 @@ variable (K) in
 lemma nnnorm_nsmul [NormedAddCommGroup E] [NormedSpace K E] (n : ℕ) (x : E) :
     ‖n • x‖₊ = n • ‖x‖₊ := by simpa [Nat.cast_smul_eq_nsmul] using nnnorm_smul (n : K) x
 
+section NormedField
+variable [NormedField E] [CharZero E] [NormedSpace K E]
+include K
+
 variable (K) in
-lemma norm_nnqsmul [NormedField E] [CharZero E] [NormedSpace K E] (q : ℚ≥0) (x : E) :
-    ‖q • x‖ = q • ‖x‖ := by simpa [NNRat.cast_smul_eq_nnqsmul] using norm_smul (q : K) x
+lemma norm_nnqsmul (q : ℚ≥0) (x : E) : ‖q • x‖ = q • ‖x‖ := by
+  simpa [NNRat.cast_smul_eq_nnqsmul] using norm_smul (q : K) x
 
 variable (K) in
-lemma nnnorm_nnqsmul [NormedField E] [CharZero E] [NormedSpace K E] (q : ℚ≥0) (x : E) :
-    ‖q • x‖₊ = q • ‖x‖₊ := by simpa [NNRat.cast_smul_eq_nnqsmul] using nnnorm_smul (q : K) x
+lemma nnnorm_nnqsmul (q : ℚ≥0) (x : E) : ‖q • x‖₊ = q • ‖x‖₊ := by
+  simpa [NNRat.cast_smul_eq_nnqsmul] using nnnorm_smul (q : K) x
+
+@[bound]
+lemma norm_expect_le {ι : Type*} {s : Finset ι} {f : ι → E} : ‖𝔼 i ∈ s, f i‖ ≤ 𝔼 i ∈ s, ‖f i‖ :=
+  Finset.le_expect_of_subadditive norm_zero norm_add_le fun _ _ ↦ by rw [norm_nnqsmul K]
+
+end NormedField
 
 theorem mul_self_norm (z : K) : ‖z‖ * ‖z‖ = normSq z := by rw [normSq_eq_def', sq]
 
diff --git a/Mathlib/Data/Complex/Basic.lean b/Mathlib/Data/Complex/Basic.lean
index 6483469eea3bc..089de7d56ab00 100644
--- a/Mathlib/Data/Complex/Basic.lean
+++ b/Mathlib/Data/Complex/Basic.lean
@@ -442,6 +442,15 @@ lemma re_ofNat (n : ℕ) [n.AtLeastTwo] : (no_index (OfNat.ofNat n) : ℂ).re =
 @[simp, norm_cast] lemma ratCast_re (q : ℚ) : (q : ℂ).re = q := rfl
 @[simp, norm_cast] lemma ratCast_im (q : ℚ) : (q : ℂ).im = 0 := rfl
 
+lemma re_nsmul (n : ℕ) (z : ℂ) : (n • z).re = n • z.re := smul_re ..
+lemma im_nsmul (n : ℕ) (z : ℂ) : (n • z).im = n • z.im := smul_im ..
+lemma re_zsmul (n : ℤ) (z : ℂ) : (n • z).re = n • z.re := smul_re ..
+lemma im_zsmul (n : ℤ) (z : ℂ) : (n • z).im = n • z.im := smul_im ..
+@[simp] lemma re_nnqsmul (q : ℚ≥0) (z : ℂ) : (q • z).re = q • z.re := smul_re ..
+@[simp] lemma im_nnqsmul (q : ℚ≥0) (z : ℂ) : (q • z).im = q • z.im := smul_im ..
+@[simp] lemma re_qsmul (q : ℚ) (z : ℂ) : (q • z).re = q • z.re := smul_re ..
+@[simp] lemma im_qsmul (q : ℚ) (z : ℂ) : (q • z).im = q • z.im := smul_im ..
+
 @[deprecated (since := "2024-04-17")]
 alias rat_cast_im := ratCast_im
 
diff --git a/Mathlib/Data/Complex/BigOperators.lean b/Mathlib/Data/Complex/BigOperators.lean
index 43c85d7750bfa..cd9cb5f344fa0 100644
--- a/Mathlib/Data/Complex/BigOperators.lean
+++ b/Mathlib/Data/Complex/BigOperators.lean
@@ -3,14 +3,15 @@ Copyright (c) 2017 Kevin Buzzard. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kevin Buzzard, Mario Carneiro
 -/
-import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Algebra.BigOperators.Expect
 import Mathlib.Data.Complex.Basic
 
 /-!
 # Finite sums and products of complex numbers
-
 -/
 
+open scoped BigOperators
+
 namespace Complex
 
 variable {α : Type*} (s : Finset α)
@@ -23,12 +24,24 @@ theorem ofReal_prod (f : α → ℝ) : ((∏ i ∈ s, f i : ℝ) : ℂ) = ∏ i
 theorem ofReal_sum (f : α → ℝ) : ((∑ i ∈ s, f i : ℝ) : ℂ) = ∑ i ∈ s, (f i : ℂ) :=
   map_sum ofReal _ _
 
+@[simp, norm_cast]
+lemma ofReal_expect (f : α → ℝ) : (𝔼 i ∈ s, f i : ℝ) = 𝔼 i ∈ s, (f i : ℂ) :=
+  map_expect ofReal ..
+
 @[simp]
 theorem re_sum (f : α → ℂ) : (∑ i ∈ s, f i).re = ∑ i ∈ s, (f i).re :=
   map_sum reAddGroupHom f s
 
+@[simp]
+lemma re_expect (f : α → ℂ) : (𝔼 i ∈ s, f i).re = 𝔼 i ∈ s, (f i).re :=
+  map_expect (LinearMap.mk reAddGroupHom.toAddHom (by simp)) f s
+
 @[simp]
 theorem im_sum (f : α → ℂ) : (∑ i ∈ s, f i).im = ∑ i ∈ s, (f i).im :=
   map_sum imAddGroupHom f s
 
+@[simp]
+lemma im_expect (f : α → ℂ) : (𝔼 i ∈ s, f i).im = 𝔼 i ∈ s, (f i).im :=
+  map_expect (LinearMap.mk imAddGroupHom.toAddHom (by simp)) f s
+
 end Complex
diff --git a/Mathlib/Data/NNReal/Basic.lean b/Mathlib/Data/NNReal/Basic.lean
index df15ccc645e66..acc5c21138221 100644
--- a/Mathlib/Data/NNReal/Basic.lean
+++ b/Mathlib/Data/NNReal/Basic.lean
@@ -3,16 +3,13 @@ Copyright (c) 2018 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Algebra.Algebra.Defs
+import Mathlib.Algebra.BigOperators.Expect
 import Mathlib.Algebra.Order.BigOperators.Ring.Finset
 import Mathlib.Algebra.Order.Field.Canonical.Basic
-import Mathlib.Algebra.Order.Nonneg.Field
 import Mathlib.Algebra.Order.Nonneg.Floor
+import Mathlib.Algebra.Ring.Regular
 import Mathlib.Data.Real.Pointwise
 import Mathlib.Order.ConditionallyCompleteLattice.Group
-import Mathlib.Tactic.Bound.Attribute
-import Mathlib.Tactic.GCongr.CoreAttrs
-import Mathlib.Algebra.Ring.Regular
 
 /-!
 # Nonnegative real numbers
@@ -55,6 +52,7 @@ This file defines `ℝ≥0` as a localized notation for `NNReal`.
 assert_not_exists Star
 
 open Function
+open scoped BigOperators
 
 -- to ensure these instances are computable
 /-- Nonnegative real numbers. -/
@@ -278,22 +276,26 @@ theorem coe_multiset_sum (s : Multiset ℝ≥0) : ((s.sum : ℝ≥0) : ℝ) = (s
 theorem coe_multiset_prod (s : Multiset ℝ≥0) : ((s.prod : ℝ≥0) : ℝ) = (s.map (↑)).prod :=
   map_multiset_prod toRealHom s
 
+variable {ι : Type*} {s : Finset ι} {f : ι → ℝ}
+
 @[simp, norm_cast]
-theorem coe_sum {α} {s : Finset α} {f : α → ℝ≥0} : ↑(∑ a ∈ s, f a) = ∑ a ∈ s, (f a : ℝ) :=
+theorem coe_sum (s : Finset ι) (f : ι → ℝ≥0) : ∑ i ∈ s, f i = ∑ i ∈ s, (f i : ℝ) :=
   map_sum toRealHom _ _
 
-theorem _root_.Real.toNNReal_sum_of_nonneg {α} {s : Finset α} {f : α → ℝ}
-    (hf : ∀ a, a ∈ s → 0 ≤ f a) :
+@[simp, norm_cast]
+lemma coe_expect (s : Finset ι) (f : ι → ℝ≥0) : 𝔼 i ∈ s, f i = 𝔼 i ∈ s, (f i : ℝ) :=
+  map_expect toRealHom ..
+
+theorem _root_.Real.toNNReal_sum_of_nonneg (hf : ∀ i ∈ s, 0 ≤ f i) :
     Real.toNNReal (∑ a ∈ s, f a) = ∑ a ∈ s, Real.toNNReal (f a) := by
   rw [← coe_inj, NNReal.coe_sum, Real.coe_toNNReal _ (Finset.sum_nonneg hf)]
   exact Finset.sum_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]
 
 @[simp, norm_cast]
-theorem coe_prod {α} {s : Finset α} {f : α → ℝ≥0} : ↑(∏ a ∈ s, f a) = ∏ a ∈ s, (f a : ℝ) :=
+theorem coe_prod (s : Finset ι) (f : ι → ℝ≥0) : ↑(∏ a ∈ s, f a) = ∏ a ∈ s, (f a : ℝ) :=
   map_prod toRealHom _ _
 
-theorem _root_.Real.toNNReal_prod_of_nonneg {α} {s : Finset α} {f : α → ℝ}
-    (hf : ∀ a, a ∈ s → 0 ≤ f a) :
+theorem _root_.Real.toNNReal_prod_of_nonneg (hf : ∀ a, a ∈ s → 0 ≤ f a) :
     Real.toNNReal (∏ a ∈ s, f a) = ∏ a ∈ s, Real.toNNReal (f a) := by
   rw [← coe_inj, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]
   exact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]
diff --git a/Mathlib/Topology/Instances/Real.lean b/Mathlib/Topology/Instances/Real.lean
index 5b895c2cdeb63..16325c2f3e0ab 100644
--- a/Mathlib/Topology/Instances/Real.lean
+++ b/Mathlib/Topology/Instances/Real.lean
@@ -3,16 +3,17 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import Mathlib.Data.Real.Star
-import Mathlib.Algebra.Algebra.Basic
+import Mathlib.Algebra.Module.Rat
+import Mathlib.Algebra.Module.Submodule.Lattice
 import Mathlib.Algebra.Periodic
+import Mathlib.Data.Real.Star
 import Mathlib.Topology.Algebra.Order.Archimedean
 import Mathlib.Topology.Algebra.Order.Field
-import Mathlib.Topology.Algebra.UniformMulAction
 import Mathlib.Topology.Algebra.Star
+import Mathlib.Topology.Algebra.UniformMulAction
 import Mathlib.Topology.Instances.Int
-import Mathlib.Topology.Order.Bornology
 import Mathlib.Topology.Metrizable.Basic
+import Mathlib.Topology.Order.Bornology
 
 /-!
 # Topological properties of ℝ