Definition essential supremum

Question

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?

Answer 1

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.