Question

Suppose that \(f\) and \(g\) are integrable on \([a, b]\) and \(f(x)=g(x)\) except for \(x\) in a set of Lebesgue measure zero. Show that $$ \int_{a}^{b} f(x) d x=\int_{a}^{b} g(x) d x $$

Short answer

**Answer:** If two functions f(x) and g(x) are equal except for x in a set with Lebesgue measure zero and both are integrable, their definite integrals from a to b will be equal.

Step-by-step solution

Define the difference of the integrals

We want to show that the integral of f(x) from a to b is equal to the integral of g(x) from a to b. To do this, we will consider their difference: $$ \int_{a}^{b} f(x) dx - \int_{a}^{b} g(x) dx $$

Combine the integrals

Using the properties of integrals, we can combine the two integrals in the above expression into one integral by taking the difference of f(x) and g(x) inside the integral: $$ \int_{a}^{b} (f(x) - g(x)) dx $$

Define the set where f(x) is not equal to g(x)

Let's denote the set where f(x) ≠ g(x) by E. By definition, the Lebesgue measure of this set is zero: \(m(E) = 0\).

Use properties of Lebesgue integration

We have $$ \int_{a}^{b} (f(x) - g(x)) dx = \int_{E} (f(x) - g(x)) dx + \int_{[a,b] \setminus E} (f(x) - g(x)) dx $$ On the set \([a,b] \setminus E\), we have f(x) = g(x), so \((f(x) - g(x)) = 0\). Therefore, the second integral in the above expression is zero: $$ \int_{[a,b] \setminus E} (f(x) - g(x)) dx = 0 $$

Focus on the integral over the set E with measure zero

Now we only need to consider the first integral in Step 4: $$ \int_{E} (f(x) - g(x)) dx $$ Since the Lebesgue measure of E is zero, it follows that the integral of any integrable function over a set with measure zero is zero. Thus, we have $$ \int_{E} (f(x) - g(x)) dx = 0 $$

Conclude the proof

We have shown that both integrals in Step 4 are zero. Therefore, the difference between the integrals of f(x) and g(x) from a to b is zero: $$ \int_{a}^{b} (f(x) - g(x)) dx = 0 $$ This concludes the proof that $$ \int_{a}^{b} f(x) dx = \int_{a}^{b} g(x) dx $$

Key concepts

Integrable Functions

In the world of mathematics, particularly within the scope of real analysis, the concept of integrable functions plays a fundamental role in understanding the behavior of functions across intervals. An integrable function is essentially one that can be integrated, meaning we can compute its integral, which is key to determining many properties such as area under the curve.
For a function to be considered integrable in the context of Lebesgue integration, it must satisfy certain conditions regarding its boundedness and the nature of its domain. Specifically, a function is Lebesgue integrable if it's measurable and the integral of its absolute value is finite; this ensures that the 'total size' of the function is contained, so to speak. An interesting aspect of Lebesgue integration is its ability to handle functions with infinite discontinuities or those that are not well-behaved in the traditional sense, as long as they meet the criteria of integrability.
This broadens the horizons beyond the capabilities of the Riemann integral, which requires functions to be quite well-behaved on any interval over which they are integrated. To further illustrate, the exercise provided deals with two functions, both of which are assumed to be integrable on a given interval. The problem demonstrates the significance of integrability, as it allows us to assert that the two functions have the same integral over the interval even if they differ by a negligible amount.

Lebesgue Measure

Imagine you're trying to describe how 'large' or 'small' a set is; this is where the Lebesgue measure comes in handy within the framework of real analysis. The Lebesgue measure is a way to assign a 'size' to sets, especially subsets of the real number line, in a manner that aligns with our intuitive understanding of length.
A set can have a Lebesgue measure of zero, which intuitively means that the set is so small, it doesn't contribute to the overall 'size'—think of it as almost like the mathematical equivalent of dust particles that don't add weight to a scale. This property is pivotal to our exercise, as the set where the functions differ has a Lebesgue measure of zero, affirming that its influence on the integral of the function is nonexistent. The exercise elegantly showcases how the concept of Lebesgue measure is integral (pardon the pun) to understanding the nuances of integration within real analysis.

Real Analysis

Nestled at the heart of higher mathematics is real analysis, a branch that deals with the behavior of real numbers, sequences, series, and functions. It provides the rigorous underpinning for calculus, which in turn is foundational to many scientific disciplines.
Real analysis grapples with concepts of limits, convergence, continuity, and, of course, integration and differentiation. Central to real analysis is the establishment of the Lebesgue integral, which has transformed the process of integration by allowing us to integrate a wider class of functions—those that are not necessarily continuous. As shown in the exercise, the principles of real analysis enable us to navigate through complex problems, such as proving the equality of integrals of functions that are not exactly the same everywhere but differ only on a set with a Lebesgue measure of zero. By reinforcing these concepts, students of real analysis can develop a deep appreciation for the precision and abstraction that characterize this area of study, ultimately contributing to its elegant applications across all branches of mathematics and beyond.