The space $S$ is the space of all smooth functions $\phi:\mathbb{R}^n\to\mathbb{R}$ such that for every (integer) $N$, $|\phi(x)|\cdot|x|^N\to 0$ as $|x|\to \infty$ (rapidly decreasing functions).
Given locally integrable function $f$ such that for all $A>0$
$\int_{|x|\leq A}|f|\leq A^M$
(for some fixed integer $M>0$)
How can I show that the integral
$\int_{\mathbb{R}^n}f\cdot \phi $
is well defined for every $\phi\in S$?
thank you.
Let $m_0>0$ such as $|\phi(x)|\leq\dfrac 1{\vert x\vert^{M+2}}$ for all $x$ such as $\vert x\vert>m_0$.
For any $m\geq m_0$, we have that $$\int_{m<\vert x\vert\leq m+1} |f(x)| |\phi(x)| dx \leq \frac{(m+1)^M}{m^{M+2}}$$ hence the convergence of $$\int_{m_0<\vert x\vert\leq m_0+n} |f(x)| |\phi(x)| dx\leq\sum_{k=0}^{n-1}\frac{(m_0+k+1)^M}{(m_0+k)^{M+2}}$$ as $n$ goes to infinity, and the result.