If the pre-image of the function is whole real line and is defined as following: f ( x ) = { 1 if x ∈ Z 0 otherwise 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.

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If the pre-image of the function is whole real line and is defined as following:  f  (  x  )  =      {                            1                          if                    x          ∈                      Z                                                0                          otherwise                        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.

Brendan Bush · Accepted Answer

Yes, indeed the essential supremum of this function is zero given the fact that union of sets of measure zero is still zero.

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

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