Let ( X , M , μ ) be a measure space with positive measure μ such that μ ( X ) = 1. Let f be an integrable function. I have been working on a proof for lim p → ∞ | | f | | L p = | | f | | L ∞ , and I have it all worked out except one piece:If | | f | | L p &lt; ∞ for all p ≥ 1, then | | f | | L ∞ &lt; ∞Here I am using the definition | | f | | L ∞ = inf { λ : | f | ≤ λ a . e . } . Any help would be greatly appreciated!

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Let   (  X  ,      M    ,  μ  ) be a measure space with positive measure   μ such that   μ  (  X  )  =  1. Let    f be an integrable function. I have been working on a proof for       lim          p      →      ∞            |        |    f      |              |                      L        p              =      |        |    f      |              |                      L        ∞            , and I have it all worked out except one piece:If       |        |    f      |              |                      L                  p                      &amp;lt;  ∞ for all   p  ≥  1, then       |        |    f      |              |                      L                  ∞                      &amp;lt;  ∞Here I am using the definition       |        |    f      |              |                      L                  ∞                      =  inf            {        λ  :      |    f      |    ≤  λ    a  .  e  .            }      . Any help would be greatly appreciated!

Leslie Rollins · Accepted Answer

This is not true. The   p-norm of   x  ↦  log  ⁡  (  1      /    x  ) on (0,1) is   (  Γ  (  p  +  1  )      )          1              /            p        &amp;lt;  ∞, but its   ∞-norm is not finite.

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