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Mathematics for Physics

227 Integrability

A property of a function under which the integral of its absolute value is finite that is used to say the function has finite total size under the integral.

definition [d] (Integrability) A property of a function \(f\): the integral of \(|f|\) is finite.

where

  • \(f\) is a function on a domain of integration.
  • \(|f|\) is the absolute value of \(f\).
  • the scalar field of values of \(f\) may be \(\mathbb{R}\).
  • the scalar field of values of \(f\) may be \(\mathbb{C}\).

Note:

  • finiteness of \(\int |f|\) is the classical absolute-integrability condition associated with \(L^{1}\).
  • under Lebesgue theory a function is integrable exactly when it is absolutely integrable.
  • the \(L^{1}\)-norm is \(\lVert f \rVert_{1} = \int |f|\, ds\).
  • Riemann integrability of a bounded function on \([a,b]\) means the limit of Riemann sums exists and is unique.
  • Riemann integrability need not pass to pointwise limits of integrable functions.

227.1 Elementary Example

227.1.1 Simple

Integrability means \(\int |f|\) is finite. On three sample heights, the discrete stand-in is a finite sum of absolute values.

\[ A = \{ 1,\ 2,\ 3 \} \]

\[ f(1) = -1,\quad f(2) = 2,\quad f(3) = -3 \]

\[ \sum_{a \in A} |f(a)| = 6 < \infty \]

where

  • \(f\) is the function.
  • the sum is a discrete model of \(\int |f|\).

227.1.2 General

On an interval, absolute integrability is finiteness of the integral of \(|f|\).

\[ f(x) = e^{-|x|}\ \text{on }\mathbb{R} \]

\[ \int_{-\infty}^{\infty} |f(x)|\, dx = 2 < \infty \]

where

  • \(\lVert f \rVert_{1} = \int |f|\, ds\) is the \(L^{1}\)-norm.

227.2 References

  1. Pietsch, A. History of Banach Spaces and Linear Operators. Birkhäuser, 2007. — integrability conditions underlying \(L^{p}\) via integrals of powers of \(|f|\).
  2. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms, 5th ed. Matrix Editions, 2015. — Lebesgue integrability as absolute integrability.
  3. Stewart, J., Clegg, D., & Watson, S. Calculus: Early Transcendentals. Cengage Learning, 2020. — Riemann integrability via Riemann sums.
  4. Gamelin, T. W., & Greene, R. E. Introduction to Topology, 2nd ed. Dover, 1999. — \(\lVert f\rVert_{1}=\int|f|\).