The Role of the Singularity in the Proof

Oct 13, 2025 by windailys.com 72 views

Introduction

Fubini's theorem is a fundamental result in real analysis that allows us to compute the integral of a function over a product space by iterated integration. The theorem states that if we have a function $f(x,y)$ defined on a product space $X \times Y$ , and if the function is integrable with respect to the product measure, then we can compute the integral of $f$ over the product space by iterated integration with respect to the individual measures on $X$ and $Y$ . However, the proof of Fubini's theorem relies heavily on the assumption that the function is integrable, and it is not immediately clear where the proof fails for non-integrable functions.

The Role of Integrability in Fubini's Theorem

In order to understand where the proof of Fubini's theorem fails for non-integrable functions, we need to examine the role of integrability in the proof. The proof of Fubini's theorem typically proceeds by showing that the function $f(x,y)$ can be approximated by a sequence of integrable functions $f_n(x,y)$ , and then using the dominated convergence theorem to show that the iterated integrals converge to the same value. However, if the function $f(x,y)$ is not integrable, then this approximation argument fails, and the proof of Fubini's theorem breaks down.

The Case of Non-Integrable Functions

So, where does the proof of Fubini's theorem fail for non-integrable functions? The answer lies in the fact that the proof relies heavily on the assumption that the function is integrable. If the function is not integrable, then the approximation argument fails, and the proof of Fubini's theorem breaks down. In particular, the proof fails when the function has a singularity at a point in the product space, or when the function has a discontinuity that is not integrable.

The Role of the Singularity in the Proof

Let's consider the case where the function $f(x,y)$ has a singularity at a point $(x_0,y_0)$ in the product space. In this case, the function is not integrable with respect to the product measure, and the proof of Fubini's theorem breaks down. The reason for this is that the approximation argument fails, and the dominated convergence theorem cannot be applied. In particular, the sequence of integrable functions $f_n(x,y)$ that are used to approximate the function $f(x,y)$ will not converge to the function $f(x,y)$ at the point $(x_0,y_0)$ , and the iterated integrals will not converge to the same value.

The Case of Discontinuous Functions

Another case where the proof of Fubini's theorem fails is when the function $f(x,y)$ has a discontinuity that is not integrable. In this case, the function is not integrable with respect to the product measure, and the proof of Fubini's theorem breaks down. The reason for this is that the approximation argument fails, and the dominated convergence theorem cannot be applied. In particular, the sequence of integrable functions $f_n(x,y)$ that are used to approximate the function $f(x,y)$ will not converge to the function $f(x,y)$ at the point of discontinuity, and the iterated integrals will not converge to the same value.

The Role of the Product Measure in the Proof

The proof of Fubini's theorem also relies heavily on the assumption that the function is integrable with respect to the product measure. If the function is not integrable with respect to the product measure, then the proof of Fubini's theorem breaks down. In particular, the dominated convergence theorem cannot be applied, and the approximation argument fails. This is because the product measure is not absolutely continuous with respect to the individual measures on $X$ and $Y$ , and the function is not integrable with respect to the product measure.

Conclusion

In conclusion, the proof of Fubini's theorem fails for non-integrable functions when the function has a singularity at a point in the product space, or when the function has a discontinuity that is not integrable. The proof also fails when the function is not integrable with respect to the product measure. The reason for this is that the approximation argument fails, and the dominated convergence theorem cannot be applied. In particular, the sequence of integrable functions that are used to approximate the function will not converge to the function at the point of singularity or discontinuity, and the iterated integrals will not converge to the same value.

References

Fubini, G. (1907). Sugli integrali multipli. Rendiconti del Circolo Matematico di Palermo, 27, 141-147.
Tonelli, L. (1909). Sulle funzioni di due variabili. Rendiconti del Circolo Matematico di Palermo, 28, 141-147.
Rudin, W. (1976). Real and complex analysis. McGraw-Hill.

Glossary

Fubini's theorem: A fundamental result in real analysis that allows us to compute the integral of a function over a product space by iterated integration.
Lebesgue integral: A generalization of the Riemann integral that is defined for functions that are not necessarily integrable with respect to the Riemann measure.
Product measure: A measure that is defined on a product space by combining the individual measures on the factors.
Dominated convergence theorem: A result in real analysis that states that if a sequence of functions converges pointwise to a function, and if the sequence is dominated by an integrable function, then the limit function is integrable.
Q&A: Where does the proof of Fubini's theorem fail for non-integrable functions? ================================================================================

Q: What is Fubini's theorem, and why is it important?

A: Fubini's theorem is a fundamental result in real analysis that allows us to compute the integral of a function over a product space by iterated integration. It is important because it provides a powerful tool for solving problems in mathematics and physics that involve multiple variables.

Q: What is the role of integrability in Fubini's theorem?

A: The proof of Fubini's theorem relies heavily on the assumption that the function is integrable with respect to the product measure. If the function is not integrable, then the proof of Fubini's theorem breaks down.

Q: What happens when the function has a singularity at a point in the product space?

A: When the function has a singularity at a point in the product space, the function is not integrable with respect to the product measure, and the proof of Fubini's theorem breaks down. The reason for this is that the approximation argument fails, and the dominated convergence theorem cannot be applied.

Q: What happens when the function has a discontinuity that is not integrable?

A: When the function has a discontinuity that is not integrable, the function is not integrable with respect to the product measure, and the proof of Fubini's theorem breaks down. The reason for this is that the approximation argument fails, and the dominated convergence theorem cannot be applied.

Q: What is the role of the product measure in the proof of Fubini's theorem?

A: The proof of Fubini's theorem also relies heavily on the assumption that the function is integrable with respect to the product measure. If the function is not integrable with respect to the product measure, then the proof of Fubini's theorem breaks down.

Q: Can you provide an example of a function that is not integrable with respect to the product measure?

A: Yes, consider the function $f(x,y) = \frac{1}{x}$ defined on the product space $\mathbb{R}^2$ . This function is not integrable with respect to the product measure because it has a singularity at the point $(0,0)$ .

Q: What are some common mistakes to avoid when applying Fubini's theorem?

A: Some common mistakes to avoid when applying Fubini's theorem include:

Assuming that the function is integrable with respect to the product measure without checking.
Failing to check for singularities or discontinuities in the function.
Not using the dominated convergence theorem to justify the interchange of limits.

Q: How can I determine whether a function is integrable with respect to the product measure?

A: To determine whether a function is integrable with respect to the product measure, you can use the following steps:

Check for singularities or discontinuities in the function.
Use the dominated convergence theorem to justify the interchange of limits.
Check that the function is integrable with respect to the individual measures on the factors.

Q: What are some common applications of Fubini's theorem?

A: Some common applications of Fubini's theorem include:

Computing the integral of a function over a product space.
Solving problems in mathematics and physics that involve multiple variables.
Using the dominated convergence theorem to justify the interchange of limits.

Q: Can you provide some additional resources for learning more about Fubini's theorem?

A: Yes, some additional resources for learning more about Fubini's theorem include:

The book "Real and Complex Analysis" by Walter Rudin.
The book "Measure Theory" by Paul Halmos.
The online resource "Fubini's Theorem" on the website MathWorld.

Q: How can I apply Fubini's theorem to solve problems in mathematics and physics?

A: To apply Fubini's theorem to solve problems in mathematics and physics, you can use the following steps:

Identify the function and the product space.
Check for singularities or discontinuities in the function.
Use the dominated convergence theorem to justify the interchange of limits.
Compute the integral of the function over the product space.

Q: What are some common challenges when applying Fubini's theorem?

A: Some common challenges when applying Fubini's theorem include:

Checking for singularities or discontinuities in the function.
Justifying the interchange of limits using the dominated convergence theorem.
Computing the integral of the function over the product space.

Q: How can I overcome these challenges?

A: To overcome these challenges, you can use the following strategies:

Use the dominated convergence theorem to justify the interchange of limits.
Check for singularities or discontinuities in the function.
Use the product measure to compute the integral of the function over the product space.

Q: What are some common mistakes to avoid when applying Fubini's theorem?

A: Some common mistakes to avoid when applying Fubini's theorem include:

Assuming that the function is integrable with respect to the product measure without checking.
Failing to check for singularities or discontinuities in the function.
Not using the dominated convergence theorem to justify the interchange of limits.