Weak ergodic averages over dilated measures

Let $m\in\mathbb{N}$ and $\textbf{X}=(X,\mathcal{X},\mu,(T_{\alpha})_{\alpha\in\mathbb{R}^{m}})$ be a measure preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\nu$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\textbf{X}$ if there exists $A\subseteq\mathbb{R}$ of density 1 such that for all $f\in L^{\infty}(\mu)$, we have $$\lim_{t\in A,t\to\infty}\int_{\mathbb{R}^{m}}f(T_{t \alpha}x)\,d\nu(\alpha)=\int_{X}f\,d\mu$$ for $\mu$-a.e. $x\in X$. Let $W(\textbf{X})$ denote the collection of all $\alpha\in\mathbb{R}^{m}$ such that the $\mathbb{R}$-action $(T_{t\alpha})_{t\in\mathbb{R}}$ is not ergodic. Under the assumption of the pointwise convergence of double Birkhoff ergodic average, we show that a Borel measure $\nu$ on $\mathbb{R}^{m}$ is weakly equidistributed for an ergodic system $\textbf{X}$ if and only if $\nu(W(\textbf{X})+\beta)=0$ for every $\beta\in\mathbb{R}^{m}$. Under the same assumption, we also show that $\nu$ is weakly equidistributed for all ergodic measure preserving systems with $\mathbb{R}^{m}$-actions if and only if $\nu(\ell)=0$ 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.