Consider a measure space ( Ω , μ ), where μ ( Ω ) &lt; ∞.It is classically known that if p &lt; q, then L q ( Ω , μ ) ⊄ L p ( Ω , μ ). The form of the argument q &gt; p ≥ 1, we construct a function f ∈ L p ( Ω , μ ) such that f ∉ L q ( Ω , μ ).Is it possible to take one such function f∈Lp(Ω,μ) such that f ∉ L p + ϵ ( Ω , μ ) for every ϵ &gt; 0?

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Consider a measure space   (  Ω  ,  μ  ), where   μ  (  Ω  )  &amp;lt;  ∞.It is classically known that if   p  &amp;lt;  q, then       L    q    (  Ω  ,  μ  )  ⊄      L    p    (  Ω  ,  μ  ). The form of the argument   q  &amp;gt;  p  ≥  1, we construct a function   f  ∈      L    p    (  Ω  ,  μ  ) such that   f  ∉      L    q    (  Ω  ,  μ  ).Is it possible to take one such function f∈Lp(Ω,μ) such that   f  ∉      L          p      +      ϵ        (  Ω  ,  μ  ) for every   ϵ  &amp;gt;  0?

Ryan Newman · Accepted Answer

A necessary and sufficient condition to find an   f  ∈            L        p   but not in             L              p      +      ε       is that the involved measure space contains sets of arbitrarily small measure.

Consider a measure space ( <mi mathvariant="normal">Ω , μ<!-- μ

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