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A continuous irreducible [[Unitary representation|unitary representation]]  $  \pi $
of a locally compact [[Unimodular group|unimodular group]]  $  G $
in a Hilbert space  $  H $
such that for some non-zero vector  $  \xi \in H $
the function  $  g \mapsto ( \pi ( g) \xi , \xi ) $, 
$  g \in G $, 
is integrable with respect to the [[Haar measure|Haar measure]] on  $  G $. 
In this case,  $  \pi $
is a square-integrable representation and there exists a dense vector subspace  $  H  ^  \prime  \subset  H $
such that  $  g \mapsto ( \pi ( g) \xi , \eta ) $, 
$  g \in G $, 
is an integrable function with respect to the Haar measure on  $  G $
for all  $  \xi , \eta \in H  ^  \prime  $. 
If  $  \{ \pi \} $, 
the unitary equivalence class of the representation  $  \pi $, 
denotes the corresponding element of the dual space  $  \widehat{G}  $
of  $  G $, 
then the singleton set containing  $  \{ \pi \} $
is both open and closed in the support  $  \widehat{G}  _ {r} $
of the [[Regular representation|regular representation]].

====Comments====
Instead of integrable representation one usually finds square-integrable representation in the literature. Let  $  \pi $
and  $  \pi  ^  \prime  $
be two square-integrable representations; then the following orthogonality relations hold:

$$ 
\int\limits _ { G } ( \pi ( g) \xi , \eta )
\overline{ {( \pi  ^  \prime  ( g) \xi  ^  \prime  , \eta  ^  \prime  ) }}\; \ 
d g =
$$

$$ 
= \ 
\left \{ 
\begin{array}{ll}
0   &\textrm{ if }
 \pi  \textrm{ and }  \pi  ^  \prime   \textrm{ are  not
 equivalent ,  }  \\
d _  \pi   ^ {-1}
( \xi , \xi  ^  \prime  ) ( \eta , \eta  ^  \prime  )   &\textrm{ if }  \pi = \pi  ^  \prime  ,  \\
\end{array}

 \right .$$

where the integral is with respect to Haar measure. The scalar  $  d _  \pi  $
is called the formal degree or formal dimension of  $  \pi $. 
It depends on the normalization of the Haar measure  $  d g $. 
If  $  G $
is compact, then every irreducible unitary representation  $  \pi $
is square integrable and finite dimensional, and if Haar measure is normalized so that  $  \int _ {G} dg = 1 $, 
then  $  d _  \pi  $
is its dimension.

The square-integrable representations are precisely the irreducible subrepresentations of the left (or right) regular representation on  $  L _ {2} ( G) $
and occur as discrete direct summands.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Warner,   "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)  pp. Sect. 4.5.9 {{ZBL|0265.22020}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S.A. Gaal,   "Linear analysis and representation theory" , Springer  (1973)  pp. Chapt. VII</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  pp. 138  (Translated from Russian)</TD></TR>
</table>