Finitely additive probabilities in the real line 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?
 A: I object to 

the sequence $\mu_r$ [converges] to a finitely additive probability measure. 

because it suggests we have a limit as $r\to\infty$. In reality, the following happens: every $\mu_r$ is a point of the unit sphere in $L_1(\mathbb R)$, which is a subset of the unit sphere of $L_\infty(\mathbb R)^*$. By weak* compactness, $\mu_r$ have at least one limit point in the unit sphere of $L_\infty(\mathbb R)^*$. (We can think of any element of $L_\infty(\mathbb R)^*$ as a finitely additive measure because it can be applied to characteristic functions of measurable set.) Any such limit point could be called $\mu$ and there are many of them. For example, on the set $A=\bigcup_k [4^k, 2\cdot 4^k]$ the measures $\mu_r$ take values anywhere between $1/6$ and $1/3$. By restricting to a subsequence with $\mu_{r_j}(A)=\gamma$ for a particular $\gamma\in (1/6,1/3)$ we get a finitely additive measure $\mu$ with $\mu(A)=\gamma$. This already gives uncountably many choices for $\mu$.
And whatever choice of  $\mu$ and $\nu$ are made, they will not be the same, as Did pointed out: $\mu((-\infty,b])=1/2$ and $\nu((-\infty,b])=0$ for any real $b$.
For more information, search for invariant mean or read about amenable groups.
