Talk:Locally integrable function

Latest comment: 1 year ago by Spireguy in topic Examples

Euclidean space

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I don't know much about this subject, which is what led me to this page. But the article gives a weaker definition than is apparently available, saying the more abstract setting is not used very often. Here is at least one reader who needed it! Maybe somebody who knows more about this subject than I do can improve the article. Preceding unsigned comment by 216.165.151.43

In this vein, here are some useful references:

  • Schilling & Kuhn (2021) and Driver (2012) define   given an arbitrary topological space   and an arbitrary measure   on the Borel  -algebra  
  • Edwards & Gaudry (1977) and Filter & Weber (1997) define   given an arbitrary measure space   a function   is in   iff for every   (i.e.,   is measurable and has finite measure),   where   is the subspace measure induced on  . [ Jacobs (1978) gives the definition for an arbitrary "measure"/" -content" on an arbitrary "local  -ring" (a.k.a.  -ring).] [Filter & Weber give a version for an arbitrary ring of sets  . At first glance, Filter & Weber may appear to say that any locally integrable function will be integrable; taking   in their definition yields  . However, their definition corresponds to our  -algebra definition only when we consider the ring   of finite-measure measurable sets. In this case, it is obvious that their definition coincides with ours.]

Thatsme314 (talk) 10:16, 9 May 2022 (UTC)Reply

Examples

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In what sense is the final example a solution to the given differential equation on all of  ? I get that the restriction of this function to   does not extend to a distribution on  , but it's unclear how the function given (with  ) can be interpreted as some kind of weak solution of the given equation across zero, since it's not integrable. Spireguy (talk) 15:05, 21 May 2023 (UTC)Reply