Let
$m\in \mathbb{N}$
and
$\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$
be a measure-preserving system with an
$\mathbb{R}^{m}$
-action. We say that a Borel measure
$\unicode[STIX]{x1D708}$
on
$\mathbb{R}^{m}$
is weakly equidistributed for
$\mathbf{X}$
if there exists
$A\subseteq \mathbb{R}$
of density 1 such that, for all
$f\in L^{\infty }(\unicode[STIX]{x1D707})$
, we have
$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$
for
$\unicode[STIX]{x1D707}$
-almost every
$x\in X$
. Let
$W(\mathbf{X})$
denote the collection of all
$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$
such that the
$\mathbb{R}$
-action
$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$
is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure
$\unicode[STIX]{x1D708}$
on
$\mathbb{R}^{m}$
is weakly equidistributed for an ergodic system
$\mathbf{X}$
if and only if
$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$
for every
$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$
. Under the same assumption, we also show that
$\unicode[STIX]{x1D708}$
is weakly equidistributed for all ergodic measure-preserving systems with
$\mathbb{R}^{m}$
-actions if and only if
$\unicode[STIX]{x1D708}(\ell )=0$
for all hyperplanes
$\ell$
of
$\mathbb{R}^{m}$
. Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic-theoretic approach.