feat(measure_theory/function/lp_space): use has_measurable_add2 inste…

…ad of second_countable_topology (#11202)

Use the weaker assumption `[has_measurable_add₂ E]` instead of `[second_countable_topology E]` in 4 lemmas.

src/measure_theory/function/lp_space.lean

@@ -971,23 +971,23 @@ begin
   exact snorm'_lt_top_of_snorm'_lt_top_of_exponent_le hfq_m hfq_lt_top (le_of_lt hp_pos) hpq_real,
 end
 
-lemma snorm'_sum_le [second_countable_topology E] {ι} {f : ι → α → E} {s : finset ι}
+section has_measurable_add
+variable [has_measurable_add₂ E]
+
+lemma snorm'_sum_le {ι} {f : ι → α → E} {s : finset ι}
   (hfs : ∀ i, i ∈ s → ae_measurable (f i) μ) (hq1 : 1 ≤ q) :
   snorm' (∑ i in s, f i) q μ ≤ ∑ i in s, snorm' (f i) q μ :=
 finset.le_sum_of_subadditive_on_pred (λ (f : α → E), snorm' f q μ)
   (λ f, ae_measurable f μ) (snorm'_zero (zero_lt_one.trans_le hq1))
   (λ f g hf hg, snorm'_add_le hf hg hq1) (λ x y, ae_measurable.add) _ hfs
 
-lemma snorm_sum_le [second_countable_topology E] {ι} {f : ι → α → E} {s : finset ι}
+lemma snorm_sum_le {ι} {f : ι → α → E} {s : finset ι}
   (hfs : ∀ i, i ∈ s → ae_measurable (f i) μ) (hp1 : 1 ≤ p) :
   snorm (∑ i in s, f i) p μ ≤ ∑ i in s, snorm (f i) p μ :=
 finset.le_sum_of_subadditive_on_pred (λ (f : α → E), snorm f p μ)
   (λ f, ae_measurable f μ) snorm_zero (λ f g hf hg, snorm_add_le hf hg hp1)
   (λ x y, ae_measurable.add) _ hfs
 
-section second_countable_topology
-variable [second_countable_topology E]
-
 lemma mem_ℒp.add {f g : α → E} (hf : mem_ℒp f p μ) (hg : mem_ℒp g p μ) : mem_ℒp (f + g) p μ :=
 ⟨ae_measurable.add hf.1 hg.1, snorm_add_lt_top hf hg⟩
 
@@ -1006,7 +1006,7 @@ begin
     exact (hf i (s.mem_insert_self i)).add (ih (λ j hj, hf j (finset.mem_insert_of_mem hj))), },
 end
 
-end second_countable_topology
+end has_measurable_add
 
 end borel_space