If the pre-image of the function is whole real line and is defined as following: \begin{equation} f(x) = \begin{cases} 1 & \text{if}\,x\in\mathbb{Z} \\ 0 & \text{otherwise} \end{cases} \end{equation}
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.
Yes, indeed the essential supremum of this function is zero given the fact that union of sets of measure zero is still zero.