Problem 51 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable, and let \(g:[a, b] \rightarrow \mathbb{R}\) be a bounded function such that the set \(\\{x \in[a, b]: g(x) \neq f(x)\\}\) is of content zero. Show that \(g\) is integrable and $$ \int_{a}^{b} g(x) d x=\int_{a}^{b} f(x) d x $$ (Compare Proposition 6.13.)

Step-by-Step Solution

Answer

In summary, to show that $g$ is integrable and $\int_{a}^{b} g(x) dx=\int_{a}^{b} f(x) dx$, we first defined the content zero set as a union of intervals, and used the boundedness of $g$ and $f$ to show that $g$ is integrable on these intervals. We then proved that the integrals of $f$ and $g$ are equal by showing that the difference between their Riemann sums is arbitrarily small. Hence, we concluded that $\int_{a}^{b} g(x) dx = \int_{a}^{b} f(x)dx$.

Step 1: Define content zero set as a union of intervals

Let \(E=\{x \in [a, b]: g(x) \neq f(x)\}\) be the content zero set mentioned in the problem. Since \(E\) has content zero, we consider a family of open intervals \((c_i, d_i)\) that cover \(E\) and satisfy \(\sum_{i=1}^\infty (d_i -c_i) < \epsilon\) for any \(\epsilon > 0\). Define \(I_i = (c_i, d_i)\) and rewrite the interval \([a, b]\) as the union of intervals, \([a, b] = \bigcup_{i=1}^\infty I_i \cup ([a, b] \setminus E)\).

Step 2: Show \(g\) is integrable using the boundedness of \(g\) and \(f\)

We'll now show that the function \(g\) is integrable by first showing that it satisfies the Riemann criterion. Let us choose a partition \(P\) of \([a, b]\) such that each \(I_i\) is contained in one of the intervals of the partition. For this partition, we know that \(f\) and \(g\) agree on every interval of the partition except for the intervals containing elements from the content zero set. Since \(g\) is bounded, we can find a partition such that the difference between upper and lower sums is arbitrarily small. Now, we can bound the difference between the Riemann sums for \(f\) and \(g\): \(\left|\sum_{i=1}^\infty (M_i - m_i)(d_i-c_i)\right| \leq \sum_{i=1}^\infty (M_i - m_i)(d_i-c_i) < \epsilon\), where \(M_i\) and \(m_i\) are the upper and lower bounds respectively for \(f\) or \(g\) on their corresponding intervals.

Step 3: Prove \(\int_a^b g(x)dx = \int_a^b f(x)dx\)

Now we'll show that the integrals of \(f\) and \(g\) are equal. We know that the integral is the limit of the Riemann sums as the norm of the partition goes to zero. From step 2, we have that the Riemann sums of \(f\) and \(g\) are arbitrarily close: \(\left|\sum_i M_i(d_i-c_i) - \sum_i m_i(d_i-c_i)\right| < \epsilon\). Hence, as the norm of the partition goes to zero, both the Riemann sums of \(f\) and \(g\) converge to their respective integrals, and we have \(\left|\int_a^b g(x) dx - \int_a^b f(x) dx\right| < \epsilon\). Since \(\epsilon\) is arbitrary, we can choose any positive number such that the difference between the integrals is smaller than that number. This implies that the integrals are equal, i.e., \(\int_a^b g(x) dx = \int_a^b f(x)dx\). We have shown that the function \(g\) is integrable and that its integral is equal to the integral of the function \(f\).

Key Concepts

Riemann Integral

The Riemann Integral is a fundamental concept in calculus. It allows us to assign a number to the area under a curve defined by a function over a closed interval \([a, b]\). To understand the Riemann integral, one needs to comprehend the process of partitioning the interval and summing up areas of small rectangles.

• Partitioning the Interval: This involves dividing the interval \([a, b]\) into smaller sub-intervals. Each sub-interval width is usually denoted as \(\Delta x_i\).
• Choosing Points: For each sub-interval, a point is chosen, usually on the left, right, or middle, to evaluate the function.
• Summing Rectangles: The function value at these points is multiplied by the interval's width to approximate the area under the curve with rectangles.

The Riemann integral \(\int_a^b f(x) \, dx\) is then the limit of these summed areas as the width of the partitions approaches zero, thus refining the approximation. This process solidifies the foundation for understanding more complex integrals.

Content Zero Set

A Content Zero Set is a set of points on the real line that, in a sense, can be "ignored" when considering integration. These sets are critical when fine-tuning the accuracy of Riemann integrals involving functions that may not align perfectly.

• Defining Content Zero: A set is of content zero if it can be covered by a collection of intervals whose total length is arbitrarily small.
• Importance in Integration: When a function differs from an integrable function only on a content zero set, it preserves the property of being integrable.

In the context of integrability, working with content zero sets often ensures that minor discrepancies in function values do not affect the integral's outcome, allowing different functions to share the same integral despite local variations.

Integrability

Integrability ensures that a function can indeed be "integrated," meaning that it is possible to measure the area under its curve. For a function to be integrable on an interval, it must meet certain conditions, prominently involving the concept of Riemann sums.

• Uniform Boundedness: The function must be bounded on the interval \([a, b]\). Both upper and lower bounds should be clearly defined.
• Convergence of Riemann Sums: The difference between the upper sum and lower sum can be made arbitrarily small as the partition becomes finer, ensuring the sums converge to a unique value.

The exercise shows that function \(g\) inherits integrability from function \(f\) by demonstrating that \(g\) only deviates from \(f\) on a content zero set. Thus, their integrals are equal, emphasizing the robustness of the integrability condition in cases with minor discrepancies.