Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric: $ d(x,y)=\sum_{i\geq 1}\frac{|x_i-y_i|}{2^i}. $

Set $\Omega_k=[-k,k]^{\mathbb{N}},$ is clearly that $\bigcup \Omega_k=\Omega$.

Denote by $B(\Omega,\mathbb{R})$ (or $B(\Omega)$ for short) the space of all bounded real functions, endowed with the sup norm (which makes $(B(\Omega)$ a Banach space). We also consider the subspaces $C_b(\Omega)$ of the bounded continuous functions ($C_b(\Omega)$ is a Banach space with the induced norm).

**My problem:** Let $(f_k)$ be a sequence of functions
in $B(\Omega)$ satisfying $f_k|\Omega_k \in C_b(\Omega_k)$ and
$f_k=0$ outside $\Omega_k$. Assume that $f_n\to f\in C_b(\Omega)$
where the above convergence is uniform in compact sets.

Now suppose that $(\nu_k)$ it is a sequence of probability measures on the Borel sigma algebra of $\Omega$ satisfying $\nu_k(f_k)=\int_{\Omega}f_kd\nu_k=1$

**My Question:** Suppose that $\nu_k\to \nu$ in the weak topology to another
probability measure $\nu$. Could I affirm that $f$ is not identically 0?