[Merged by Bors] - feat(analysis/mean_inequalities, measure_theory/lp_space): extend mem_Lp.add to all p in ennreal #5828
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…_Lp.add to all p in ennreal (#5828) Show `(a ^ q + b ^ q) ^ (1/q) ≤ (a ^ p + b ^ p) ^ (1/p)` for `a,b : ennreal` and `0 < p <= q`. Use it to show that for `p <= 1`, if measurable functions `f` and `g` are in Lp, `f+g` is also in Lp (the case `1 <= p` is already done).
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…l inequalities (#10836) Make `nnreal` versions of the existing `ennreal` lemmas ```lean lemma add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p ``` and similar, introduced by @RemyDegenne in #5828. Refactor the proofs of the `ennreal` versions to pass through these.
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Show
(a ^ q + b ^ q) ^ (1/q) ≤ (a ^ p + b ^ p) ^ (1/p)fora,b : ennrealand0 < p <= q.Use it to show that for
p <= 1, if measurable functionsfandgare in Lp,f+gis also in Lp (the case1 <= pis already done).