An 
-space on a measure
space 
, for a real number 
with 
,
consists of equivalence classes of measurable
functions 
, where 
or 
, whose 
th powers of absolute
values are integrable. Two functions
are identified when they agree almost everywhere
with respect to 
(Axler 2026, pp. 202-203). This generalizes the L2-space
of square integrable functions.
The 
norm is
 |
(1)
|
The 
-space is a Banach
space (Axler 2026, p. 205). For 
, its norm
does not arise from an inner product when the measure space contains two disjoint
sets of finite positive measure.
A one-point measure space
whose point has finite positive measure instead gives a one-dimensional Hilbert
space for every such 
.
The 
-space on 
with Lebesgue measure
is the completion of the continuous
functions with compact support using the 
norm. It
is possible for a sequence of functions 
to converge
in 
but not in 
for some other 
. For example, 
converges
in 
to 
,
which is not in 
. Although every 
is in 
, the sequence
cannot converge there. If a sequence converges in both 
and 
on the same measure space, its limits
agree almost everywhere (Axler 2026, p. 205).
For 
, the dual
normed space to 
is identified with 
, where 
.
More precisely, every bounded linear
functional is uniquely of the form
 |
(2)
|
where 
and 
(Axler 2026, p. 275). The integral
exists by Hölder's inequality. In particular,
is self-dual.
For complex functions, the inner
product form 
of this identification is antilinear (Axler 2026,
pp. 233-234). On degenerate measure spaces,
other 
-spaces can also be self-dual,
as in the one-dimensional example above.
While the use of 
functions is not as common as 
,
they are very important in analysis and partial
differential equations. For instance, some operators
are only bounded in 
for some 
.
For 
, the same integrability condition and integral
expression define an 
quasi-norm, which in general need not satisfy the triangle inequality. For any 
,
the weak 
space consists of measurable functions with
 |
(3)
|
This is a quasi-norm, and 
is contained in weak 
(Grafakos 2014, pp. 2-6). The distinction is important
for endpoint bounds such as that for the bilinear
Hilbert transform.
