Let ( Ω , μ ) be a σ-finite measure space. Suppose 1 ≤ p &lt; ∞. Consider the cone L p ( Ω ) + of positive functions of L p ( Ω ).Is L p ( Ω ) + weak-closed in L p ( Ω ) ?

Question

Let   (  Ω  ,  μ  ) be a   σ-finite measure space. Suppose   1  ≤  p  &amp;lt;  ∞. Consider the cone       L    p    (  Ω      )    +   of positive functions of       L    p    (  Ω  ).Is       L    p    (  Ω      )    +   weak-closed in       L    p    (  Ω  ) ?

nifeonibonitozg · Accepted Answer

For   p  &amp;gt;  1,       L    p   is reflexive and this set is convex so it is weakly closed because it is norm-closed (Mazur&#039;s lemma).In       L    1  , suppose that   (      f    n        )          n      =      1        ∞   is a sequence of non-negative functions converging weakly to   f. Suppose that   f is strictly negative on a set   A of positive measure. Let   g be the indicator function of   A. We have      ∫    Ω        f    n    g  =      ∫    A        f    n    →      ∫    Ω    f  g  =      ∫    A    f  &amp;lt;  0.This is impossible as       f    n   are non-negative. This argument will work in       L    p   too, so Mazur&#039;s lemma is not quite necessary.

Let ( <mi mathvariant="normal">Ω , μ ) be a

Answered question

Answer & Explanation

New Questions in Pre-Algebra