Set of discontinuties with measure 0

In summary, the function f(x,y) = 1 if x not equal to 1/2, 1 if x = 1/2 and y rational, and 0 if x = 1/2 and y irrational is an example given by the professor in class for Fubini's Theorem. The professor stated that this function is Riemann integrable because the set of discontinuities D = {x = 1/2} has measure 0. However, the student initially had doubts about this statement due to the fact that the function is discontinuous at all points of the form (1/2,y) where y is an irrational number in [0,1]. However, upon further discussion,
  • #1
Buri
273
0
My professor gave this function in class as an example (for Fubini's Theorem)

f : [0,1]² -> R defined as:

f(x,y) = 1 if x not equal to 1/2, 1 if x = 1/2 and y rational, and 0 if x = 1/2 and y irrational.

My professor said that this function is Riemann integrable since the set of discontinuities D = {x = 1/2} has measure 0. However, I don't see why this is true. The function is discontinuous at all points of the form (1/2,y) where y is an irrational number in [0,1], but the irrationals in [0,1] are uncountable so I couldn't possibly cover it with countably many rectangles with total volume less than epsilon. Maybe it was meant to be y rational?

Thanks for the help!
 
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  • #2
Wait never mind...the set of discontinuities is actually x = 1/2 and all y in [0,1] which is a line (and that has measure 0). Sometimes just asking a question helps me because I had just left my computer and realized where I was going wrong lol
 

1. What does it mean for a set of discontinuities to have measure 0?

Having a measure of 0 means that the set of discontinuities has no length, width, or height. In other words, the set takes up no space in the given metric or space.

2. How does a set of discontinuities with measure 0 differ from a set with non-zero measure?

A set with measure 0 can still contain an infinite number of points, while a set with non-zero measure contains a finite number of points. Additionally, a set with measure 0 has no width or height, while a set with non-zero measure has a measurable length, width, or height.

3. Can a set of discontinuities have measure 0 in all dimensions?

Yes, a set of discontinuities can have measure 0 in all dimensions. This means that the set takes up no space in any direction.

4. How is the measure of a set of discontinuities with measure 0 calculated?

The measure of a set of discontinuities with measure 0 is calculated using the concept of Lebesgue measure, which is based on the number of points in the set and their distribution within the given metric or space.

5. What are some real-world examples of sets of discontinuities with measure 0?

Some real-world examples of sets of discontinuities with measure 0 include the set of rational numbers, the set of points where a function is not continuous, and the Cantor set. These sets have no width or length, but still contain an infinite number of points.

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