Riemann integral and Lebesgue integral

https://iwant2study.org/lookangejss/math/Calculus/ejss_model_e_Lebesgue_integral/e_Lebesgue_integral_Simulation.xhtml

1. Overview:

This document provides a briefing on a JavaScript simulation applet designed to visually and interactively demonstrate the concepts of the Riemann integral and the Lebesgue integral. The applet is part of the Open Educational Resources / Open Source Physics @ Singapore initiative, aimed at enhancing the learning and teaching of mathematics (specifically calculus) through simulations. The tool allows users to compare and contrast these two fundamental approaches to integration using the example of a parabola, y = x².

2. Main Themes and Important Ideas:

• Comparative Visualization of Riemann and Lebesgue Integrals: The core function of the applet is to provide a side-by-side visual representation of how the Riemann and Lebesgue integrals approximate the area under a curve.
• Riemann Integral: The applet illustrates the Riemann integral using the concept of "nesting" the area between the x-axis and the function y = f(x) between upper and lower approximation sums. These sums are constructed from series of rectangles with bases along the x-axis. The upper sum uses the supremum (largest value) of y in each interval as the height, while the lower sum uses the infimum (smallest value). The text explicitly states: "In two dimensions the Rieman integral determines the area between the x axis and the function y = f(x) by nesting it between two approximation sums. Both are constructed by a series of rectangles with intervals of the integration range along the x axis. For the upper sum the approximative value of y in each interval is equal to the largest value in the interval (its supremum); for the lower sum it is equal to the smallest value (infimum)."
• Lebesgue Integral: In contrast, the Lebesgue integral divides the integration area into "stripes" parallel to the x-axis (intervals along the y-axis). The size of each stripe's area is defined by its "Lebesgue measure," which is taken parallel to the x-axis. The text explains: "The Lebesgue întegral divides the integration area into stripes (not necessarily of the same width) parallel to the x axis (intervals along the y axis). The size of the area of each interval is characterized by a measure (taken parallel to the x axis), called the Lebesgue measure."
• The simulation displays a "Riemann infimum series" in the left window and a "Lebesgue series" in the right window, using the parabola y = x² as an example.
• Interactive Exploration: The applet provides several interactive elements for users to manipulate and observe the effects on the Riemann and Lebesgue approximations:
• Draggable Endpoint (x2): Users can drag a magenta line to adjust the upper limit of the integration range.
• Sliders:A slider controls the initial point of integration (x1).
• Another slider adjusts the number of intervals (n-1 for Riemann, n for Lebesgue).
• Reset Button: Resets the integration range to 1 < x < 4 and the number of intervals to n = 10.
• Convergence and Accuracy: The simulation highlights the convergence behavior of the Riemann integral and the specific behavior of the chosen Lebesgue measure:
• For the Riemann integral, the text notes: "In the Riemann construction the approximation gets better and better with increasing n." This demonstrates the fundamental concept of the Riemann integral converging to the true area as the width of the intervals decreases.
• For the Lebesgue integral in this specific example, the applet demonstrates a "perfect coincidence" with the analytic solution "independent of the number of intervals." This is attributed to the "special definition of the Lebesgue measure chosen as an example," which was calculated analytically. The text clarifies: "With the special definition of the Lebesgue measure chosen as an example, the coincidence is perfect independent of the number of intervals (the measure was calculated analytically; it could also be calculated in a limiting series process)." It is important to note that the applet's example uses a pre-calculated Lebesgue measure for illustrative purposes, and the concept of the Lebesgue integral itself doesn't inherently provide a specific calculation method for these measures.
• Strength and Generality of the Lebesgue Integral: The source emphasizes the broader applicability of the Lebesgue integral: "The strength of the Lebesgue concept lies in the very general mathematical applicability to the concept of integration. It can be applied to any set of objects, for which a measure is defined." This highlights that while Riemann integration is well-suited for continuous functions, Lebesgue integration extends the concept of integration to a wider class of functions and spaces.
• Relationship Between Riemann and Lebesgue Integrability: The text states a crucial theoretical relationship: "A function that is Riemann integrable is always Lebesgue integrable, but not vice versa." This indicates that the Lebesgue integral is a more general concept of integration.
• Illustrative Example (Parabola): The choice of the parabola y = x² serves as a concrete example for comparing the two integration methods. The applet also displays the analytic solution y = (x³ - x1³) / 3 (yellow curve), allowing for direct comparison with the approximations.
• Experimental Activities: The "Experiments" section suggests specific tasks for users to engage with the simulation to deepen their understanding:
• E1: Comparing the Riemann infimum construction with the Lebesgue construction at default settings and observing systematic deviations.
• E2: Comparing the Riemann lower sum with a numerical rectangle algorithm and understanding why they are identical for the parabola (monotonic function) but would differ for an oscillating function like the sine function. "Why are both identical in the parabola example, while they would be different for the sine function? (the parabola increases monotonically, while the sine oscillates)."
• E3: Observing the Riemann lower sum approaching the analytic solution as the number of intervals increases and noting the difference from the Lebesgue approximation (in this specific example).
• Pedagogical Focus: The applet is presented as a tool for "Learning and Teaching Mathematics using Simulations." The inclusion of "Sample Learning Goals" (though the text is "[text]") and a "For Teachers" section underscores its educational purpose. The instructions for using the sliders and draggable boxes further support its use in a learning environment.

3. Key Facts:

• The simulation is an HTML5 applet, making it accessible through web browsers without the need for additional plugins.
• It visually compares the Riemann and Lebesgue integrals in two dimensions.
• The example function used is the parabola y = x².
• Users can interactively adjust the integration range and the number of intervals.
• The applet displays the Riemann lower sum and a specific implementation of the Lebesgue integral.
• It shows the analytic solution for comparison.
• The project is part of the Open Educational Resources / Open Source Physics @ Singapore.
• Credits are given to Dieter Roess, Fremont Teng, and Loo Kang Wee for their contributions.
• The content is licensed under the Creative Commons Attribution-Share Alike 4.0 Singapore License.

4. Conclusion:

The Riemann Integral and Lebesgue Integral JavaScript Simulation Applet provides a valuable interactive tool for students and educators to gain a deeper understanding of these fundamental concepts in calculus. By visualizing the different approaches to approximating the area under a curve and allowing for user manipulation, the applet effectively highlights the definitions, behaviors, and relative strengths of the Riemann and Lebesgue integrals. The inclusion of guided experiments further enhances its pedagogical value. The applet's accessibility as an HTML5 application makes it readily usable in various learning environments.

What is the fundamental idea behind the Riemann integral for calculating the area under a curve?

The Riemann integral approximates the area under a curve by dividing the area into a series of narrow rectangles with bases along the x-axis. For each interval on the x-axis, the height of the rectangle is determined by a value of the function within that interval. The Riemann integral exists if the sums of the areas of these rectangles, calculated using both the supremum (largest value) and the infimum (smallest value) of the function in each interval, converge to the same limit as the width of the intervals decreases.

2. How does the Lebesgue integral differ conceptually from the Riemann integral in calculating the area under a curve?

Unlike the Riemann integral, which divides the area under the curve into vertical strips (rectangles with bases on the x-axis), the Lebesgue integral divides the area into horizontal stripes (intervals along the y-axis). The area of each stripe is determined by its "Lebesgue measure," which is the length of the set of x-values for which the function's value falls within that y-interval. The Lebesgue integral is the sum of these measures multiplied by the corresponding y-values.

3. What is the significance of the "Lebesgue measure" in the context of the Lebesgue integral?

The Lebesgue measure is a way of assigning a "size" or "length" to sets of real numbers. In the context of the Lebesgue integral, it quantifies the extent (along the x-axis) of the set of points where the function takes on values within a particular range on the y-axis. If each horizontal stripe has a unique Lebesgue measure, the sum of these measures (weighted by the height of the stripe) constitutes the Lebesgue integral.

4. The provided text mentions a JavaScript simulation using the example of a parabola y = x². How does this simulation illustrate the difference between Riemann and Lebesgue integrals?

The simulation visually demonstrates the approximation process for both integrals. The left window shows the Riemann integral using the infimum (lower sum), where rectangles are formed based on x-axis intervals and the minimum y-value in each interval. The right window shows the Lebesgue integral, where horizontal stripes are formed based on y-axis intervals, and a red line (Lebesgue measure) indicates the effective width of each stripe such that the rectangle's area equals the stripe's area (even with a curved end). The simulation allows users to observe how increasing the number of intervals affects the Riemann approximation and highlights that for a specific definition of the Lebesgue measure in this example, the Lebesgue approximation can be perfect regardless of the number of intervals.

5. According to the text, what is a key advantage of the Lebesgue integral over the Riemann integral in terms of mathematical applicability?

The strength of the Lebesgue integral lies in its very general mathematical applicability. It can be applied to a broader range of functions and to integration over any set of objects for which a measure can be defined, not just intervals on the real line. The text explicitly states that a function that is Riemann integrable is always Lebesgue integrable, but the converse is not necessarily true.

6. Experiment E2 asks to compare the Riemann lower sum with a numerical rectangle algorithm starting at the initial point of the interval. Why are they identical for the parabola y = x² but would differ for the sine function?

For a monotonically increasing function like y = x² over a given interval, the infimum (smallest value) within that interval will always occur at the starting point of the interval. Therefore, the Riemann lower sum (using the infimum as the height) will be identical to a numerical rectangle algorithm that uses the function's value at the initial point of each interval as its height. However, for an oscillating function like the sine function, the infimum within an interval may not necessarily occur at the start of the interval, leading to a difference between the Riemann lower sum and the rectangle algorithm starting at the initial point.

7. Experiment E3 suggests increasing the number of intervals in the Riemann construction. What observation should be made regarding the approximation of the analytic solution, and how does this contrast with the Lebesgue approximation in the simulation's example?

As the number of intervals (n) increases in the Riemann construction, the Riemann lower sum should get closer and closer to the analytic solution (the exact value of the integral). This is because the approximation by rectangles becomes finer, reducing the "gaps" or "overestimations." In contrast, for the specific definition of the Lebesgue measure chosen in the simulation's example, the Lebesgue approximation achieves a perfect coincidence with the analytic solution independent of the number of intervals. This illustrates a fundamental difference in how the two types of integrals approach the concept of area calculation and convergence.

8. What does the text imply about the calculation of the Lebesgue measure itself? Does the Lebesgue integral concept provide a specific method for this calculation?

The text indicates that the concept of the Lebesgue integral does not inherently provide a specific method for calculating its measures. In the simulation's example, the Lebesgue measure was calculated analytically. It also mentions that it could potentially be calculated through a limiting series process. This suggests that determining the Lebesgue measure for a given function and set requires separate mathematical techniques depending on the specific problem.