Let $\lambda_r$ be the Lebesgue measure restricted to the interval $[-r,r]\subset \mathbb{R}$. Each $\lambda_r$ can be normalized to $\mu_r=\lambda_r/2r$ which is a probability. The sequence $\lambda_r$ converges for $r \rightarrow \infty$ to the Lebesgue measure on the real line, but the sequence $\mu_r$ does not converge to a countably additive measure but to a finitely additive probability measure. The limit measure $\mu$ in fact satisfies $\mu(A)=0$ for any bounded subset of $\mathbb{R}$ but $\mu(\mathbb{R})=1$.
I can obtain another finitely additive probability measure by considering the sequence $\nu_r=\dfrac{1}{e^r}\int_{-\infty}^r e^x dx$, which is a probability and at the limit satisfies again $\nu(A)=0$ for any bounded subset of $\mathbb{R}$ and $\nu(\mathbb{R})=1$.
There are infinite ways to do that. My question do the two finitely additive probability measures obtained as limits of the two different sequence $\mu_r$ and $\nu_r$ are different? In which subsets of $\mathbb{R}$ do they differ? Do you know any reference that discusses the generation of finitely additive probability measures as a limit of sequences of countably additive measures?