# Definition essential supremum

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

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

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.
• And the $t > 0$ vs $t \geq 0?$
• 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. Aug 8 '20 at 6:24