Question

If \(f\) is Lebesgue integrable on \((0, a)\) and \(g(x)=\int_{x}^{a} t^{-1} f(t) d t\), then \(g\) is integrable on \((0, a)\) and \(\int_{0}^{a} g(x) d x=\int_{0}^{a} f(x) d x\).

Short answer

\(g(x)\) is integrable on \((0, a)\), and \(\int_{0}^{a} g(x) \, dx = \int_{0}^{a} f(x) \, dx\).

Step-by-step solution

Understand the Problem

We need to show that the function \(g(x) = \int_{x}^{a} t^{-1} f(t) \, dt\) is integrable on the interval \((0, a)\) and that \(\int_{0}^{a} g(x) \, dx = \int_{0}^{a} f(x) \, dx\). Since \(f\) is Lebesgue integrable, it implies that \(\int_{0}^{a} |f(x)| \, dx < \infty\). We need to use known theorems from real analysis, such as Fubini's theorem, to achieve this.

Use Fubini's Theorem

Apply Fubini's theorem, which allows the interchange of order of integration when \(f(x)\) and \(t^{-1}\) are integrable on \((0, a)\). We write:\[ \int_{0}^{a} g(x) \, dx = \int_{0}^{a} \left( \int_{x}^{a} t^{-1} f(t) \, dt \right) \, dx. \]By Fubini's theorem, this is equal to\[ \int_{0}^{a} \left( \int_{0}^{t} dx \right) t^{-1} f(t) \, dt = \int_{0}^{a} t^{-1} f(t) \, t \, dt. \]

Simplify the Integral

In the expression \(\int_{0}^{a} t^{-1} f(t) \, t \, dt\), simplify \(t^{-1} f(t) \, t = f(t)\). Thus, the integral becomes:\[ \int_{0}^{a} f(t) \, dt. \]This shows that \(\int_{0}^{a} g(x) \, dx = \int_{0}^{a} f(x) \, dx\).

Confirm Integrability of g

Since \(g(x)\) is defined as an integral of a Lebesgue integrable function \(f(t)\) weighted by \(t^{-1}\), incorporation of Fubini's theorem and manipulation leads to the conclusion that \(g(x)\) is also integrable on \((0, a)\) due to the valid operation of integration performed.

Key concepts

Fubini's Theorem

Fubini's Theorem is a powerful result in real analysis that allows us to interchange the order of integration in certain multiple integral problems. This theorem plays a crucial role in the solution of the exercise, where we aim to show the equality of two integrals.

To apply Fubini's Theorem, the function we are dealing with must be integrable. In the given problem, both the function \(f(t)\) and the term \(t^{-1}\) are mentioned as integrable over the interval \((0, a)\). Fubini's Theorem allows us to rewrite the integral \(\int_{0}^{a} \left( \int_{x}^{a} t^{-1} f(t) \, dt \right) \, dx\) by swapping the order of integration.

• This theorem simplifies our task by turning a double integration problem into a single integral that is easier to evaluate.
• The result is \(\int_{0}^{a} f(t) \, dt\), showing the equality between different forms of integration.

Thanks to Fubini's theorem, we can better manage complex integration processes, especially when handling functions defined on multidimensional spaces.

Real Analysis

Real Analysis is a branch of mathematics that deals with the rigorous study of real numbers and real-valued functions. It's fundamentally about understanding the behavior and properties of functions, sequences, series, and sets.

In our context, Real Analysis provides the tools needed to understand when and why we can perform certain operations on integrals. Understanding the properties of Lebesgue integration and the conditions for applying theorems like Fubini's are central to solving our exercise.

• Integration Techniques: Real Analysis equips us with various techniques for integrating functions that appear odd or difficult at first.
• Convergence and Limits: Understanding the behavior of functions as they approach limits is crucial in defining and working with integrals.

Real Analysis is what allows us to rigorously justify the transitions and operations we perform on functions, ensuring that they are mathematically valid and sound.

Lebesgue Integrable Functions

Lebesgue Integrable Functions are those that can be integrated using the method developed by Henri Lebesgue. This integration approach extends the possibilities offered by classical Riemann integration, accommodating a broader class of functions.

The function \(f\) mentioned in the exercise must be Lebesgue integrable. This means that the integral of its absolute value is finite, expressed as \(\int_{0}^{a} |f(x)| \, dx < \infty\).

• Why Use Lebesgue Integration: It provides more flexibility and is better suited for limit processes involving functional series or transformations.
• Practical Usage: Many real-world applications, such as probability theory, rely on the Lebesgue approach due to its robustness.

In this problem, the fact that \(f\) is Lebesgue integrable assures us of our ability to perform the required integrations and apply relevant theorems effectively.