ABSTRACT
The characteristic function of a set A is the function OA(z) which is equal to 1 when z E A and is equal to 0 when z fl. A. The characteristic function ll[o,oo)(z) of the semiaxis z ?: 0 is called the Heaviside unit function and is denoted O(z) (Fig 2a):
0.3. The Lebesgue integral of a function f over an open set 0 is given as
The collection of all (complex-valued, measurable) functions f specified on 0 for which the norm
Iff E £P(O') for any 0'@ 0, then f is said to be p-locally integrable in 0 (for p = 1, we say it is locally integrable in 0). The collection of p-locally integrable functions in 0 is denoted .Cfoc(O), .Cfoc(lRn) = .Cfoc·
A measurable function is said to have compact support in 0 if it vanishes almost everywhere outside a certain 0' @ 0. The set of all functions in £P(O) that have compact support in 0 is denoted .C~(O).
