The discussion revolves around the existence of a continuous, strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero, specifically in the context of measure theory and the Cantor function.
Some participants have proposed modifications to the Cantor function, such as defining a new function based on it. There is ongoing exploration of the implications of these modifications on the measures of the sets involved. Multiple interpretations of the properties of the Cantor function and its modifications are being discussed.
Participants are considering the implications of the Cantor function's properties, including its continuity and the measure of its image. There is a focus on how to construct a function that meets the problem's requirements while addressing the constraints of measure theory.
sbashrawi said:Cantor function is amapping from [0,1] onto [0,1] , I need a restriction so that its image has measure zero. But it is seemed to me that its image is countable am I right?
sbashrawi said:We have the function : h(x) = C(x) + x , where C(x) is the cantor function, the function h(x) is increasing function from [0,1] onto [0,2].
[0,1] = C union O : c = cantor set , O its complement.
m( C) = 0 , m(O) = 1 .
how can I modify this to get the required function?
sbashrawi said:m( h(O)) = 1 and m(h(c))= 1 .
I think we can use the inverse function of h . it is contniuous , increasing
and hinv: [0,2] to [0,1]
and h inv(h(c)) = c which has measure zero i.e it maps a set with positive measure to a set of measure zero as required.
let f(x) = h(x) / 2 we get the required function: f: [0,1] to [0,1]