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 $n \mapsto |\phi(ng)|^2$.