Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:

If $U$ is an open neighborhood of a point $x \in G$, then there exists $f \in \mathcal{B}^1$ with $f(x)\ne 0$ and $\operatorname{supp}(f)\subseteq U$. This is supposed to be a consequence of the following theorem:

Recall that $\mathcal{B}^1:= \operatorname{span}(\mathcal{P}(G)\cap C_c(G))\cap L^1(G)$, where $\mathcal{P}(G)$ are the continuous functions of positive type on $G$.

How does this Urysohn-like result follow from the above result?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:

If $U$ is an open neighborhood of a point $x \in G$, then there exists $f \in \mathcal{B}^1$ with $f(x)\ne 0$ and $\operatorname{supp}(f)\subseteq U$. This is supposed to be a consequence of the following theorem:

Recall that $\mathcal{B}^1:= [\operatorname{span}\mathcal{P}(G)]\cap L^1(G)$, where $\mathcal{P}(G)$ are the continuous functions of positive type on $G$.

How does this Urysohn-like result follow from the above result?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:

If $U$ is an open neighborhood of a point $x \in G$, then there exists $f \in \mathcal{B}^1$ with $f(x)\ne 0$ and $\operatorname{supp}(f)\subseteq U$. This is supposed to be a consequence of the following theorem:

Recall that $\mathcal{B}^1:= \operatorname{span}(\mathcal{P}(G)\cap C_c(G))$, where $\mathcal{P}(G)$ are the continuous functions of positive type on $G$.

How does this Urysohn-like result follow from the above result?

Let $G$ be a locally compact Hausdorff group. In the proof of theorem 4.32 of Folland's book "A course in abstract harmonic analysis", the following result is used:

If $U$ is an open neighborhood of a point $x \in G$, then there exists $f \in \mathcal{B}^1$ with $f(x)\ne 0$ and $\operatorname{supp}(f)\subseteq U$. This is supposed to be a consequence of the following theorem:

Recall that $\mathcal{B}^1:= \operatorname{span}(\mathcal{P}(G)\cap C_c(G))\cap L^1(G)$, where $\mathcal{P}(G)$ are the continuous functions of positive type on $G$.

How does this Urysohn-like result follow from the above result?