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Square Integrable

A square integrable function on a measure space (X,A,mu) is a measurable function f:X->R or C satisfying

int_X|f|^2dmu<infty.

The equivalence classes of these functions, identified when equal almost everywhere, form the L2-space L^2(X,mu) (Axler 2026, pp. 194 and 202).

For Lebesgue measure on R, this condition is

int_(-infty)^infty|f(x)|^2dx<infty.

It does not imply that f is Lebesgue integrable. For example, f(x)=1/(1+|x|) is square integrable on R, but int_(-infty)^infty|f(x)|dx=infty.

See also

Integrable, L2-Norm, L2-Space, Titchmarsh Theorem

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References

Axler, S. Measure, Integration & Real Analysis. Author-hosted revision, June 12, 2026. https://measure.axler.net/MIRA.pdf.Sansone, G. "Square Integrable Functions." §1.1 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 1-2, 1991.

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Square Integrable

Cite this as:

Weisstein, Eric W. "Square Integrable." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SquareIntegrable.html

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