Proposition 8

There exists a filter basis $\mathfrak{B}$ on $A(G)\cap \mathscr{K}_+(G)$ such that the following conditions are satisfied:

(i) For every element $\varphi$ of a set of $\mathfrak{B}$, one has $\|\varphi \|_1= 1$ and $\|\mathscr{F}(\varphi )\|_{\infty}\leqslant 1$ ;

(ii) One has

lim$_{\varphi ,\mathfrak{B}}\varphi \cdot dx=\varepsilon_e$

in the space $\mathscr{C}'(G)$ of measures with compact support on G endowed with the topology of uniform convergence on compact parts of $\mathscr{C}(G)$ ;

(iii) One has

lim$_{\varphi ,\mathfrak{B}}\mathscr{F}(\varphi ) = 1$

for the topology of compact convergence on $\widehat{G}$;

(iv) For $p= 1$ or $p= 2$, and for every $f\in L^p(G)$, one has $\varphi *f\in$ A(G) for every $\varphi$ belonging to a set of $\mathfrak{B}$ and

lim$_{\varphi ,\mathfrak{B}}\varphi *f=f$

in $L^p(G)$.

Let $K_0$ be a fixed compact neighbourhood of $e$ in G. Let $\mathfrak{B}_0$ be a basis of the filter of neighbourhoods of $e$ in G consisting of symmetric compact neighbourhoods contained in $K_0$ (cf. TG, III, p. 4). For K $\in \mathfrak{B}_0$, let $X'_K$ be the set of functions $\psi \in \mathscr{K}_+(G)$ such that Supp$(\psi )\subset K$ and $\int\psi (x)dx= 1$ ; it is nonempty (lemma 1 of II, p. 200). Let $X_K$ be the set of functions $\psi *\psi$ for $\psi \in X'_K$. It is nonempty and contained in $A(G)\cap \mathscr{K}_+(G)$. The set $\mathfrak{B}$ whose elements are the sets $X_K$ for K varying in $\mathfrak{B}_0$ is a filter basis on $A(G)\cap \mathscr{K}_+(G)$. Let us prove that $\mathfrak{B}$ satisfies the required properties.

If $X\in \mathfrak{B}$ and $\varphi \in X$, one has $\|\varphi \|_1=\int_G\varphi (x)dx= 1$, whence $\|\mathscr{F}(\varphi )\|_{\infty}\leqslant 1$, which establishes property (i).

Property (ii) follows from INT, VIII, § 2, No.$^o7$, Corollary 1 to Lemma 4. A compact subset of $\widehat{G}$ is a compact subset of $\mathscr{C}(G)$, hence (ii) implies lim$_{\varphi ,\mathfrak{B}}\mathscr{F}(\varphi ) = 1$ for the topology of compact convergence on $\widehat{G}$, that is, (iii).

Finally, let $p= 1$ or $p= 2$. Let $f\in L^p(G)$. We have $\varphi *f\rightarrow f$ in $L^p(G)$ with respect to the filter $\mathfrak{B}$ (INT, VIII, §4, n$^o7$, prop. 20). Moreover, for every K in $\mathfrak{B}_0$ and $\varphi \in X_K$, there exists $\psi \in X'_K$ such that $\varphi =\psi *\psi$, whence $\varphi *f=\psi *(\psi *f)$. We have $\psi \in L^1(G)\cap L^2(G)$ and $\psi *f\in L^1(G)\cap L^2(G)$, hence $\varphi *f\in A(G)$.