I've encountered something called the "essential supremum" while working with L p spaces (in particular, for p = ∞).I tried looking it up on the internet but all the definitions use concepts from Measure Theory, which I'm not familiar with. Is there a way to wrap my head around it without having to deal with measures? I really only need to understand this to show that a piecewise continuous function f ∈ C ( Ω ) is in L ∞ ( Ω ). I can't seem to understand how the L ∞ space works or how it is defined.

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I&#039;ve encountered something called the &quot;essential supremum&quot; while working with       L          p       spaces (in particular, for   p  =  ∞).I tried looking it up on the internet but all the definitions use concepts from Measure Theory, which I&#039;m not familiar with. Is there a way to wrap my head around it without having to deal with measures? I really only need to understand this to show that a piecewise continuous function   f  ∈  C  (  Ω  ) is in       L          ∞        (  Ω  ). I can&#039;t seem to understand how the       L    ∞   space works or how it is defined.

Dustin Durham · Accepted Answer

You cannot avoid specifying a measure when talking about       L          ∞        (  Ω  ) since the space depends critically on the measure you are using. Let   (  Ω  ,      F    ,  μ  ) be a measure space. The       L          ∞       norm of a measurable function   f is defined as  ‖  f      ‖                  L                  ∞                      =  inf  {  M  ≥  0  :      |    f  (  x  )      |    ≤  M       for           μ        -almost every     x  ∈  Ω  }  .One can show that       |    f  (  x  )      |    ≤  ‖  f      ‖                  L                  ∞                     for   μ-almost every   x  ∈  Ω, i.e. the inf is a actually minimum.       L          ∞        (  Ω  ,  μ  ) is defined as the space of (equivalence classes of a.e. equal) measurable functions   f  :  Ω  →      C   such that   ‖  f      ‖                  L                  ∞                      &amp;lt;  ∞. For your purpose, I guess you are using   μ  = Lebesgue measure, so that is the only measure you need to care about. To show that   ‖  f      ‖                  L                  ∞                      &amp;lt;  ∞, you need to show that there exists   M  ≥  0 such that       |    f  (  x  )      |    ≤  M for almost every   x  ∈  Ω. So the procedure is very similar to the procedure for showing that   f is bounded, except here you are allowed exceptions on a set of measure 0. Note that for continuous functions   f and   μ a Borel measure (such as Lebesgue measure), if   μ assigns positive measure to all open subsets of   Ω, then it can be shown that   ‖  f      ‖                  L                  ∞                      =      sup          x      ∈      Ω            |    f  (  x  )      |  . This is the case when   Ω is open, or when   Ω is an interval with more than one point.

I've encountered something called the "essential supremum" while working with L <mrow

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