If the pre-image of the function is whole real line and is defined as following:

$f(x)=\{\begin{array}{ll}1& \text{if}\phantom{\rule{thinmathspace}{0ex}}x\in \mathbb{Z}\\ 0& \text{otherwise}\end{array}$

What would be the essential supremum?

I understand that the essential supremum of a function is the smallest value that is larger or equal than the function values almost everywhere when allowing for ignoring what the function does at a set of points of measure zero.

Would it be still zero considering each individual integer essentially has measure zero? However, the measure of the integer set with value 1 is not zero, is it?

Thanks in advance.

$f(x)=\{\begin{array}{ll}1& \text{if}\phantom{\rule{thinmathspace}{0ex}}x\in \mathbb{Z}\\ 0& \text{otherwise}\end{array}$

What would be the essential supremum?

I understand that the essential supremum of a function is the smallest value that is larger or equal than the function values almost everywhere when allowing for ignoring what the function does at a set of points of measure zero.

Would it be still zero considering each individual integer essentially has measure zero? However, the measure of the integer set with value 1 is not zero, is it?

Thanks in advance.