  In this paper, we first introduce a new function space $MH^{\theta, p}$ whose
norm is given by the $\ell^p$-sum of modulated $H^\theta$-norms of a given
function. In particular, when $\theta < -\frac 12$, we show that the space
$MH^{\theta, p}$ agrees with the modulation space $M^{2, p}(\mathbb R)$ on the
real line and the Fourier-Lebesgue space $\mathcal F L^{p}(\mathbb T)$ on the
circle. We use this equivalence of the norms and the Galilean symmetry to adapt
the conserved quantities constructed by Killip-Vi\c{s}an-Zhang to the
modulation space setting. By applying the scaling symmetry, we then prove
global well-posedness of the one-dimensional cubic nonlinear Schr\"odinger
equation (NLS) in almost critical spaces. More precisely, we show that the
cubic NLS on $\mathbb R$ is globally well-posed in $M^{2, p}(\mathbb R)$ for
any $p < \infty$, while the renormalized cubic NLS on $\mathbb T$ is globally
well-posed in $\mathcal FL^p(\mathbb T)$ for any $p < \infty$.
  In Appendix, we also establish analogous global-in-time bounds for the
modified KdV equation (mKdV) in the modulation spaces on the real line and in
the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the
proof in this case is a Galilean transform which converts the mKdV to the
mKdV-NLS equation.
