Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem | Introduction to Functional Analysis | Mathematics | MIT OpenCourseWare
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Lecture 12: Lebesgue Integrable Functions, the Lebesgue Integral and the Dominated Convergence Theorem | Introduction to Functional Analysis | Mathematics | MIT
MIT OpenCourseWare is a web based publication of virtually all MIT course content. OCW is open and available to the world and is a permanent MIT activity
4 months ago
Lebesgue Integral Explained
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Lebesgue rectangles are a simple geometric way to visualize how the Lebesgue integral builds up area by looking at slices of the range rather than slices of the domain. Instead of cutting the x-axis into intervals and stacking vertical rectangles, the Lebesgue approach groups together all points where the function takes values within a certain band. For each horizontal “slice” of height Δy, you look at the set of x-values where the function falls inside that slice, and then measure the total len
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MathxMatrix on Instagram: "The Lebesgue integral: a smarter way to calculate the area under a curve. While the traditional Riemann integral (what we learn in school) breaks area into vertical strips, the Lebesgue integral breaks it into horizontal slabs. For "nice" functions, they give the same answer. But Lebesgue is more powerful-using Measure Theory to handle "pathological" or jagged functions where traditional calculus fails. Original creation by @abidebyreason (also on YT)"
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