Proposition 5

Let $p\in [1,+\infty [$. The space $\mathscr{L}_{\pi}^p(G)$ is the space of functions $f\in \mathscr{F}_{\pi}(G)$ such that $f$ is $\mu$-measurable and the function $\|f\|$ belong to $\mathscr{L}^p(G/H)$.

Let $f\in \mathscr{L}_{\pi}^p(G)$. It is the limit in $\mathscr{L}_{\pi}^p(G)$ of a sequence of elements of $\mathscr{K}_{\pi}(G)$. By prop. 3, b) and Egoroff's theorem (INT, IV, p. 175, § 5, No.$^o4$, th. 2), the function $f$ is therefore $\mu$-measurable; consequently, the function $\|f\|$ on $G/H$ is $\nu$-measurable (INT, VII, p. 47, § 2, No.$^o3$, prop. 6, b)). Since $N_p(f)$ is finite, the function $\|f\|$ belong to $\mathscr{L}^p(G/H)$ (INT, IV, p. 184, § 5, No.$^o6$, th. 5).

Let us prove the converse assertion. Let $f$ be a $\mu$-measurable function belonging to $\mathscr{F}_{\pi}(G)$ such that $\|f\| \in \mathscr{L}^p(G/H)$. Let $\varepsilon >0$. Let us prove that there exists $\widetilde{f}\in \mathscr{K}_{\pi}(G)$ such that

which will conclude the proof.

Suppose first that there exists a compact subset C of G such that $f$ is zero outside $C\cdot H$. Let $u\in \mathscr{K}_+(G)$ be a function such that $u(x)>0$ for all $x\in C$ and define $v=u^1$ as above. Let us denote by $\varphi$ the characteristic function of the support of $u$.

Let $q$ be the conjugate exponent of $p$. If $p= 1$, let $w$ be the constant function on G equal to sup$_{x\in G}u(x)$. If $p >1$, define

for every $x\in G$. In both cases, the function $w$ is continuous and positive; it belongs to $\mathscr{K}_1(G)$ (Prop. 4 applied to the trivial representation of dimension 1 and to the function $u^q$), hence it is bounded on G. Put W = sup$_{x\in G}w(x)$.

Let $s$ be the function on G with values in E defined by Lemma 5 applied to $f$. Put $g=\varphi s$. The function $g$ is $\mu$-measurable, and satisfies $\|g\|\leqslant \|f\|/$ inf$_{x\in C\cdot H}v(x)$. Since $g$ vanishes outside the support of $u$ and $\kappa$ is continuous, we have $g\in \mathscr{L}_E^p(G, \kappa \cdot \mu)$. Let $\widetilde{g}$ be a function in $\mathscr{K}_E(G)$ such that

$_G$ W

We have $us=ug$, hence $f= (us)^{\pi}= (ug)^{\pi}$ almost everywhere modulo H (Lemma 5, b)). Let $\widetilde{f}= (u\widetilde{g})^{\pi}$; we have $\widetilde{f}\in \mathscr{K}_{\pi}(G)$ (Prop. 4, b)). For every $x\in G$ - $S_{ug}$, we obtain

whence

by Hölder's inequality in the case $p >1$. Since $S_{ug}$ is negligible modulo H, it follows that

by INT, VII, p. 46, § 2, n$^o3$, Prop. 5, b), which is applicable since $g-\widetilde{g}$ vanishes outside a compact subset of G. This implies the required property when $f$ vanishes outside a compact subset modulo H.

Consider now the general case. Since $\|f\| \in \mathscr{L}^p(G/H)$ by hypothesis, there exists a compact subset L of $G/H$ such that

$(G/H)-L$ 2

(cf. INT, IV, p. 152, § 4, n$^o6$, th. 4). Let $\varphi_L$ be the characteristic function of L and put $f_L= (\varphi_L\circ \varpi )f$. We have

$G/H(G/H)-L$ 2

The function $f_L$ is $\mu$-measurable and vanishes outside a compact set modulo H. It belongs to $\mathscr{L}_{\pi}^p$(G), hence by the preceding case, there exists $\widetilde{f}\in \mathscr{K}_{\pi}(G)$ such that $N_p(f_L-\widetilde{f})\leqslant (\frac{\varepsilon}{2})^{1/p}$, whence

as desired.

Let us consider the case $p= 2$. For $f_1$ and $f_2$ in $\mathscr{F}_{\pi}$(G), the function $x\mapsto \langle f_1(x)|f_2(x)\rangle$ belongs to $\mathscr{F}_1(G)$ since the representation $\pi$ is unitary, and therefore defines by passing to the quotient a function on $G/H$, which is identified as above with $\langle f_1|f_2\rangle$. One has the majorization $|\langle f_1|f_2\rangle |\leqslant \|f_1\|\|f_2\|$ in $\mathscr{F}(G/H)$.

If $f_1$ and $f_2$ belong to $\mathscr{L}_{\pi}^2$(G), then the function $\langle f_1|f_2\rangle$ belongs to $\mathscr{L}_1^1(G)$. In particular, it is integrable on $G/H$. The mapping

is a positive Hermitian form on $\mathscr{L}_{\pi}^2(G)$; the associated seminorm is the seminorm $N_2$. In particular, the space $\mathscr{L}_{\pi}^2(G)$ is a prehilbertian space and $L^2_{\pi}(G)$ is the Hilbert space associated with $\mathscr{L}_{\pi}^2(G)$.