Definition of cusp form in $L^2$ and convergence over $N_{\mathbb Q} \backslash N_{\mathbb A}$ Let $G$ be an adjoint semisimple group over $\mathbb Q$ with parabolic subgroup $P = MN$ in good position relative to a compact subgroup $U= \prod\limits_v K_v$ of $G(\mathbb A)$.  Let $L$ be the space of square integrable functions on $G(\mathbb Q) \backslash G(\mathbb A)$ which are invariant on the right by $U$.  In Euler Products, Langlands defines a cusp form in $L$ to be an element $\phi$ satisfying
$$\int\limits_{N(\mathbb Q) \backslash N(\mathbb A)} \phi(ng)dn = 0\tag{1}$$
for almost all $g \in G(\mathbb A)$.  However, it is not clear to me why the left hand side converges at all.  All we know is that 
$$\int\limits_{G(\mathbb Q) \backslash G(\mathbb A)} |\phi(g)|^2 dg < \infty$$
Using the Iwasawa decomposition, we can write $G(\mathbb A) = N(\mathbb A)M(\mathbb A)K$, so that, at least formally,
$$\int\limits_{G(\mathbb A)} \phi(g)dg = \int\limits_{M(\mathbb A)} \int\limits_{N(\mathbb A)} \int\limits_K \phi(nmk) \delta_P(m) dk dm dn = \operatorname{vol}(K)\int\limits_{M(\mathbb A)} \int\limits_{N(\mathbb A)}\phi(nm) \delta_P(m)dndm$$
One can probably finagle from here something like:
$$\int\limits_{G(\mathbb Q) \backslash G(\mathbb A)}|\phi(g)|^2 dg = \operatorname{vol}(K)\int\limits_{M(\mathbb Q)\backslash M(\mathbb A)} \int\limits_{N(\mathbb Q) \backslash N(\mathbb A)} |\phi(nm)|^2 \delta_P(m)dn dm$$
from which we should have
$$\int\limits_{N(\mathbb Q) \backslash N(\mathbb A)} |\phi(ng)|^2 dn < \infty$$
for almost all $g \in G$.  However, this says nothing about the convergence of $n \mapsto \phi(ng)$, only of $n \mapsto |\phi(ng)|^2$.
 A: I think its convergence is proven by the following method;
claim 1
Let $G$ be a 2nd countable locally compact topological group and $\Gamma$ be a discrete subgroup.
Also, let $A$ be a measureble set of $G$ and $B$ be a measurable set of $\Gamma \backslash G$ such that projection of $A$ contains $B$.
Then there exists a Borel measurable subset $A'$ of $A$ such that $A'$ projects $B$ one-to-one onto.
The proof is trivial.
claim 2
Let $f$ be the element of $L^1_{loc}$(G($\mathbb{Q}$)\G($\mathbb{A}$)), then its constant term $f_P$ along parabolic P is in $L^1_{loc}$(U($\mathbb{A}$)M($\mathbb{Q}$)\G($\mathbb{A}$)).
Sketch of the proof
We may assume that $f$ is positive.
Take a compact set $C$ in U($\mathbb{A}$)M($\mathbb{Q}$)\G($\mathbb{A}$).
By Fubini theorem for quotient measures,
\begin{align}
\int_{U(\mathbb{A})M(\mathbb{Q})\backslash G(\mathbb{A})} \chi_{C}(g) \int _{U(\mathbb{Q}) \backslash U(\mathbb{A})}f(ug) du dg 
& = \int_{P(\mathbb{Q}) \backslash G(\mathbb{A})}f(g)\chi_{C}(g)dg \\ 
& =\int_{G(\mathbb{Q})\backslash G(\mathbb{A})} f(g) \int _{P(\mathbb{Q}) \backslash G(\mathbb{Q})} \chi_{C}(\gamma g)d\gamma dg \\
\end{align}
Let $C_1$ be $G(\mathbb{Q})\backslash G(\mathbb{Q})C$, then it is compact in G($\mathbb{Q}$)\G($\mathbb{A}$) and if the above integrand is not zero, then $g$ in $C_1$.
Moreover, we can take the measurable set $C_2$ in G($\mathbb{A}$), such that $C_2$ projects $C_1$ and $C_2$ is relatively compact by claim 1.
Hence, the above formula equals
\begin{align}
\int_{G(\mathbb{Q})\backslash G(\mathbb{A})} \chi_{C_{1}}(g)f(g) \int _{P(\mathbb{Q}) \backslash G(\mathbb{Q})} \chi_{C}(\gamma g)d\gamma dg 
& = \int_{C_{2}} f(g) \int _{P(\mathbb{Q}) \backslash G(\mathbb{Q})} \chi_{C}(\gamma g)d\gamma dg \\
& = \sharp \{\gamma \in  P(\mathbb{Q}) \backslash G(\mathbb{Q}) : \gamma \in CC_{2}^{-1} \} \|f\|_{C_1}
\end{align}
The term $\sharp \{\gamma \in  P(\mathbb{Q}) \backslash G(\mathbb{Q}) : \gamma \in CC_{2}^{-1} \}$ essentially depends only on $C$ and is finite because discrete compact set is finite. Hence the claim follows.
(I have not seen the complete proof of this fact and think this proof by myself. So it may contain some mistakes...)
A: You are entirely correct that there are several analytical issues with a naive presentation of this... although the naive presentation does present the intention, which is the most important thing.
So, yes, if we're trying to make "constant term (map)" a reasonable thing, we surely want to specify what space it maps from, and what to, and surely these (vector) spaces of functions should have topologies (or, depending on taste, bornologies...) to make the constant term map(s) continuous.
I hesitate to inject here my own "personal" precisification of this, but/so anyone who wants to see one way to make it all precise (without tooooo much silliness, I hope) can see my CUP book, with a legal version at http://www.math.umn.edu/~garrett/m/v/current_version.pdf
