This problem is regarding the space of probability measures.
For $N \geq 1$, let $\{e_i^N(.), i\geq 1\}$ denote a complete orthonormal basis for $L_2[0,N]$. Let $\{f_j\}$ be countable dense in the unit ball of $C(S)$. Then it is true that $\int f_jd\nu_1 = \int f_jd\nu_2 \forall j$ implies $\nu_1=\nu_2$. Then define the metric
$$d(\nu_1(.),\nu_2(.))=\sum_{N\geq1}\sum_{i\geq1}\sum_{j\geq1}2^{-(N+i+j)} \left|\int_{0}^{N}e_i^N(t)\int f_jd\nu_1(t)dt - \int_{0}^{N}e_i^N(t)\int f_jd\nu(t)dt\right| $$
Now given that
$$\frac{1}{m}\sum_{k=1}^{m}\int f_jd\nu_{n}(t) \to \int f_jd\nu*(t)$$ a.e. $t$
One paper claims that $d(\nu_{n}(.), \nu*(.)\to 0$.
I have no clue how can one conclude convergence of the original sequence from the convergence in ceasaro mean