Fubini's Theorem: Integral Existence for Non-Continuous Functions

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In summary, the conversation discusses the application of Fubini's theorem in relation to the continuity of a function and the existence of an integral. It is stated that according to Fubini's theorem, the order of integration in a double integral can be interchanged if the integrand is continuous. However, it is clarified that the integral can still exist even if the integrand is not continuous, as long as the set of discontinuities is a measure-0 set. The use of LateX images is also mentioned as a tool for assistance in understanding the concepts being discussed.
  • #1
Pearce_09
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hello,
I posted a question a while ago about fubini's theorem, and i believe i have found my answer.. but to clear things up, i have one more question.

If a function f(x) is not continuous then the integral (by fubinis) does not exist.
is this correct. ( I believe it is correct, but i would feel better if someone else agreed as well)
thank you!
adam
 
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  • #2
IIRC, a (dumbed down) version of Fubini in analysis says that the order of integration in a double integral may be interchanged if the integrand is a continuous function. That doesn't mean the integral doesn't exist when if the integrand is not continuous.

The LateX images in your other post are not showing, so we can't help you there.
 
  • #3
A function is integrable iff the set of discontinuities is a measure-0 set. Fubini's theorem has little to do with this - it's about double integration.
 

FAQ: Fubini's Theorem: Integral Existence for Non-Continuous Functions

What is Fubini's Theorem and why is it important in mathematics?

Fubini's Theorem is a fundamental result in mathematics that provides a way to evaluate integrals of functions defined on a product space. It is important because it allows for the calculation of integrals of non-continuous functions and has many applications in areas such as probability, physics, and engineering.

What is the statement of Fubini's Theorem?

Fubini's Theorem states that if a function f(x,y) defined on a product space [a,b] x [c,d] satisfies certain conditions, then the double integral of f(x,y) over [a,b] x [c,d] can be calculated as an iterated integral, with the inner integral being with respect to x and the outer integral being with respect to y.

Can Fubini's Theorem be applied to all functions?

No, Fubini's Theorem can only be applied to functions that satisfy certain conditions, such as being integrable on the product space. It cannot be applied to functions that are not defined on a product space or do not meet the necessary criteria.

What are some examples of how Fubini's Theorem is used in real-world problems?

Fubini's Theorem has many applications in various fields. For example, it is used in calculating probabilities in statistics, finding areas and volumes in geometry, and solving differential equations in physics and engineering.

Is there a limit to the number of dimensions that Fubini's Theorem can be applied to?

Fubini's Theorem can be extended to higher dimensions, such as triple integrals in three-dimensional space. However, the conditions for its application become more complex as the number of dimensions increases, making it more difficult to use in practice.

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