Riemann → Lebesgue → L2: what “the whole point” is

• The step from Riemann to Lebesgue integration is about changing the way we “sum” a function: Riemann slices the domain into intervals and adds rectangles, while Lebesgue slices the range into levels and measures how large the preimages of those levels are, which makes limits and convergence behave far better and integrates many more functions.[1][2]
• Once Lebesgue integration is in place, Lp spaces—especially L2—become natural objects: L2 is the space of square-integrable functions (modulo a.e. equality), and it is a Hilbert space where inner products, orthogonality, Parseval/Plancherel, and Fourier analysis live cleanly.[2][1]
• Conceptually: Riemann is the classical area-adding picture; Lebesgue is the measure-theoretic completion that stabilizes limits; L2 is the quadratic-energy viewpoint that turns functions into vectors with geometry, enabling powerful tools like orthogonal expansions and projections.[3][1][2]

Why move beyond Riemann?

• Convergence theorems: Lebesgue’s Monotone and Dominated Convergence theorems give robust conditions to pass limits through integrals—something that is fragile or unavailable in the Riemann setting.[4][2]
• More integrable functions: Many functions that fail to be Riemann integrable (e.g., with dense discontinuities) are integrable in the Lebesgue sense because measurability and “a.e.” control are the right criteria.[5][2]
• Higher-dimensional and abstract settings: Lebesgue extends cleanly to general measure spaces, which is essential for probability, ergodic theory, and harmonic analysis.[6][2]

Why L2 is “the destination” for analysis

• Inner product structure: L2(Ω) has ⟨f,g⟩ = ∫ f ḡ, making it a complete Hilbert space where orthogonal projections, best approximations, and basis expansions (e.g., Fourier) are well-defined and converge in norm.[3][2]
• Fourier theory: Plancherel’s theorem identifies the Fourier transform as an isometry on L2, so energy is preserved and series/transforms are controlled by the L2 norm, not delicate pointwise properties.[2][3]
• Riesz–Fischer: Square-summable Fourier coefficients correspond exactly to L2 functions, tying series and functions via completeness of L2.[3][2]

The throughline in one breath

Riemann gives a geometric sum for nice functions; Lebesgue rebuilds integration via measure so limits and irregular functions are handled correctly; then L2 packages “square-integrable” functions into a Hilbert space so orthogonality, projection, and Fourier analysis work seamlessly—this is the practical arc from classical calculus to modern analysis.[1][2][3]

If helpful, can give a concrete example that is Lebesgue integrable but not Riemann integrable, and show how its L2 norm is computed and used in Fourier analysis.[5][2][3]

[1] https://www.ams.org/dol/051

[2] https://www.taylorfrancis.com/books/9781482229530

[3] http://choicereviews.org/review/10.5860/CHOICE.38-2786

[4] https://arxiv.org/pdf/1702.04236.pdf

[5] https://arxiv.org/pdf/1504.04765.pdf

[6] http://arxiv.org/pdf/2104.05256.pdf

[7] https://www.research-publication.com/amsj/all-issues/vol-09/iss-07

[8] http://www.jbe-platform.com/content/journals/10.1075/lab.4.4.04gom

[9] https://www.degruyter.com/document/doi/10.1051/978-2-7598-0322-4/html

[10] http://www.compadre.org/per/items/detail.cfm?ID=13151

[11] http://link.springer.com/10.1007/978-0-8176-8232-3_10

[12] https://www.taylorfrancis.com/books/9781000714166

[13] http://link.springer.com/10.1007/978-3-0348-7448-9

[14] https://arxiv.org/pdf/2212.12566.pdf

[15] https://arxiv.org/pdf/0802.4076.pdf

[16] https://tns.ewapublishing.org/media/ea5e27811d6e4a2db6fd6194c45c0aef.marked.pdf

[17] https://arxiv.org/pdf/2502.19388.pdf

[18] https://arxiv.org/abs/2209.03637

[19] https://www.mdpi.com/2227-7390/8/12/2250/pdf

[20] http://arxiv.org/pdf/2407.08144.pdf