 # If the pre-image of the function is whole real line and is defined as following: f ( x Lucia Grimes 2022-07-05 Answered
If the pre-image of the function is whole real line and is defined as following:
$f\left(x\right)=\left\{\begin{array}{ll}1& \text{if}\phantom{\rule{thinmathspace}{0ex}}x\in \mathbb{Z}\\ 0& \text{otherwise}\end{array}$
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?
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Yes, indeed the essential supremum of this function is zero given the fact that union of sets of measure zero is still zero.