Integrable function

An integrable function has a finite integral of its absolute value over its domain. This condition makes it an element of the space of absolutely integrable functions, usually denoted L¹.

Lebesgue integral: A measure-based definition of integration that extends the Riemann integral to broad classes of functions. Absolute integrability is defined using this integral in the standard measure-theoretic setting.
L1 norm: The integral of a function's absolute value, defining its norm in L¹. Its finite value is precisely the quantitative test for integrability.
Fourier transform: A transform expressing a function in terms of its frequency components. For L¹ functions, its defining integral converges absolutely at every frequency.
Square-integrable function: A measurable function whose squared absolute value has finite integral. Square integrability differs from absolute integrability, though finite-measure domains link the two classes.
Completeness (functional analysis): The property that every Cauchy sequence in a metric space converges within that space. L¹ is complete, so Cauchy sequences of integrable functions have integrable limits in norm.
Measure space: A set equipped with a sigma-algebra and a measure assigning sizes to measurable subsets. The domain's measure determines which functions have finite absolute integral.
Simple function: A measurable function taking only finitely many values. Integrable functions can be approximated by simple functions in the L¹ norm.
Convolution: An operation combining two functions by integrating products of shifted values. Integrable inputs support convolution estimates and many regularity results.
Locally integrable function: A function integrable on every compact subset of its domain. Local integrability permits infinite total integral over an unbounded domain.
Approximate identity: A family of kernels whose convolutions approach the identity operation. Convolution with an approximate identity converges to the original function in L¹.

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