Lebesgue integration over sets of measure zero

In summary, given a measure space (X, \mathcal M, \mu) with \mu a positive measure, \mu(X) = 1, and E \in \mathcal M satisfying \mu(E) = 0, if f is Lebesgue integrable, then \int_E f d\mu = 0. This can be shown by bounding the integral with \mu(E) and using 0 \cdot \infty = 0 in measure theory.
  • #1
AxiomOfChoice
533
1
Is it true in general that if [itex]f[/itex] is Lebesgue integrable in a measure space [itex](X,\mathcal M,\mu)[/itex] with [itex]\mu[/itex] a positive measure, [itex]\mu(X) = 1[/itex], and [itex]E \in \mathcal M[/itex] satisfies [itex]\mu(E) = 0[/itex], then

[tex]
\int_E f d\mu = 0
[/tex]

This is one of those things I "knew" to be true yesterday, and the day before, and the day before...but now I can't show it! I need to be able to bound that integral, somehow, by [itex]\mu(E)[/itex], but how? Using Holder's inequality? But don't I need to know that [itex]f\in L^2[/itex] or [itex]f\in L^\infty[/itex] to do that? Do I know either of those? I don't think so...
 
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  • #2
It's fairly easy, but it's tricky. Basically, we put

[tex]\int_E fd\mu\leq \int \|f\|_\infty d\mu=\|f\|_\infty \int d\mu=\|f\|_\infty \mu(E)[/tex]

Note that [itex]\|f\|_\infty=+\infty[/itex] can happen here, that's the tricky part. The thing isd that [itex]0.(+\infty)[/itex] is defined as 0 in measure theory.
 
  • #3
micromass said:
It's fairly easy, but it's tricky. Basically, we put

[tex]\int_E fd\mu\leq \int \|f\|_\infty d\mu=\|f\|_\infty \int d\mu=\|f\|_\infty \mu(E)[/tex]

Note that [itex]\|f\|_\infty=+\infty[/itex] can happen here, that's the tricky part. The thing isd that [itex]0.(+\infty)[/itex] is defined as 0 in measure theory.

Ah yes! [itex]0 \cdot \infty = 0[/itex]! I had forgotten about that. Thanks micromass.
 

Related to Lebesgue integration over sets of measure zero

1. What is Lebesgue integration over sets of measure zero?

Lebesgue integration over sets of measure zero is a mathematical concept used in measure theory, which is a branch of mathematics that deals with the concept of size or "measure" of sets. It involves integrating (finding the area under the curve) over sets of points that have a measure of zero, which means they have no length, area, or volume.

2. Why is Lebesgue integration over sets of measure zero important?

Lebesgue integration over sets of measure zero is important because it allows us to integrate over a larger class of functions than the traditional Riemann integral. This is especially useful in cases where the traditional Riemann integral fails, such as with functions that are discontinuous or unbounded. It also has applications in probability theory and other areas of mathematics.

3. How is Lebesgue integration over sets of measure zero different from the Riemann integral?

The main difference between Lebesgue integration over sets of measure zero and the Riemann integral is the concept of "approximating" the integral. In the Riemann integral, the area under the curve is approximated by rectangles, while in Lebesgue integration, the area is approximated by a sum of the values of the function at certain points. This allows for a wider class of functions to be integrated.

4. What is the significance of sets of measure zero in Lebesgue integration?

Sets of measure zero are significant in Lebesgue integration because they allow for the integration of functions that are not continuous, and have "holes" or "gaps" in their graphs. These sets can be thought of as "negligible" or having no impact on the overall integral, but they still need to be taken into account in order to accurately calculate the integral over the entire domain.

5. Are there any limitations to Lebesgue integration over sets of measure zero?

While Lebesgue integration over sets of measure zero is a powerful tool, there are still some limitations. It is not always possible to integrate every function using this method, and there are still some functions that are not Lebesgue integrable. Additionally, the concept of "sets of measure zero" can be difficult to visualize and understand, leading to potential challenges in applying this method in certain situations.

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