International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 30   | 1 | 2 |

## Section 1.3.2.3.6. Distributions associated to locally integrable functions

G. Bricognea

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#### 1.3.2.3.6. Distributions associated to locally integrable functions

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Let f be a complex-valued function over Ω such that exists for any given compact K in Ω; f is then called locally integrable.

The linear mapping from to defined bymay then be shown to be continuous over . It thus defines a distribution :As the continuity of only requires that , is actually a Radon measure.

It can be shown that two locally integrable functions f and g define the same distribution, i.e.if and only if they are equal almost everywhere. The classes of locally integrable functions modulo this equivalence form a vector space denoted ; each element of may therefore be identified with the distribution defined by any one of its representatives f.