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Consider the following fragment from Axler's "Measure, Integration & Real analysis":

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Is the definition of $\Vert f \Vert_\infty$ unaffected when we change any of the strict inequalities into non-strict inequalities?

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If $\mu \{x:|f(x)| >t\}=0$ then $\mu \{x:|f(x)| \geq t+\epsilon\}=0$ for any $\epsilon >0$ and it follow from this that the infima are the same whether you have greater than or greater than or equal to in the definition.

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  • $\begingroup$ And the $t > 0$ vs $t \geq 0?$ $\endgroup$
    – user745578
    Aug 7 '20 at 10:20
  • $\begingroup$ Also, is the infinum a minimum? $\endgroup$
    – user745578
    Aug 7 '20 at 10:21
  • $\begingroup$ No, the infimum need not be a minimum. $\endgroup$ Aug 7 '20 at 10:23
  • $\begingroup$ I disagree with the statement that $>$ cannot be changed to $\ge$ in both places. It is fine to use $\ge$ in both places, and the quantity $||f||_\infty$ does not change if one makes that change. $\endgroup$ Aug 8 '20 at 6:24
  • $\begingroup$ @SheldonAxler You are right. That was a mistake. $\endgroup$ Aug 8 '20 at 6:31

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