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I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 [here][1] for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

EDIT. As the OP pointed out, the result is stated for compactly supported functions. By an approximation argument and the last line of the quoted theorem, an extension seems possible to functions $f:G\to\mathbb{C}$ satisfying $\int_G |f(g)|\ |\pi(g)v|_\mu\ dg < \infty$ for each seminorm $|\cdot|_\mu$ that participates in the definition of the topology of $V$.

I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 here for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

EDIT. As the OP pointed out, the result is stated for compactly supported functions. By an approximation argument and the last line of the quoted theorem, an extension seems possible to functions $f:G\to\mathbb{C}$ satisfying $\int_G |f(g)|\ |\pi(g)v|_\mu\ dg < \infty$ for each seminorm $|\cdot|_\mu$ that participates in the definition of the topology of $V$.

I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 here for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

EDIT. As the OP pointed out, the result is stated for compactly supported functions. I think an extension to integrable functions is straightforward when $V$ is a complete metric space like a Frechet space. Indeed, one can choose a sequence $f_n\in C_c^\infty(G)$ converging to $f$ in $L^1(G)$. The last line in the quoted theorem implies that the vectors $w_n:=\pi(f_n)v\in V$ form a Cauchy sequence by $\mathrm{dist}(w_n,w_m)\leq 2|f_n-f_m|_{L^1(G)}$. Now by completeness $(w_n)$ converges to some $w\in V$ and this $w$ would be $\pi(f)v$ by definition.

I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 [here][1] for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

EDIT. As the OP pointed out, the result is stated for compactly supported functions. By an approximation argument and the last line of the quoted theorem, an extension seems possible to functions $f:G\to\mathbb{C}$ satisfying $\int_G |f(g)|\ |\pi(g)v|_\mu\ dg < \infty$ for each seminorm $|\cdot|_\mu$ that participates in the definition of the topology of $V$.

I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 here for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 here for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

EDIT. As the OP pointed out, the result is stated for compactly supported functions. I think an extension to integrable functions is straightforward when $V$ is a complete metric space like a Frechet space. Indeed, one can choose a sequence $f_n\in C_c^\infty(G)$ converging to $f$ in $L^1(G)$. The last line in the quoted theorem implies that the vectors $w_n:=\pi(f_n)v\in V$ form a Cauchy sequence by $\mathrm{dist}(w_n,w_m)\leq 2|f_n-f_m|_{L^1(G)}$. Now by completeness $(w_n)$ converges to some $w\in V$ and this $w$ would be $\pi(f)v$ by definition.