Suppose we have a sample $x_1, x_2, \ldots, x_{n_1}$ of size $n_1$ from a Gaussian distribution $N(\mu_1,\sigma_1^2)$ and another sample $y_1, y_2, \ldots, y_{n_2}$ of size $n_2$ from $N(\mu_2,\sigma_2^2)$ (i.e., we have labeled observations from class 1 and class 2). Now choose some point $u$ uniformly at random over a given interval $[a,b]\subset\mathbb{R}$. Can we compute the probability (or tight bounds) that the closest observation to $u$ is from (say) class 1?
In an attempt to formulate the question, as a first step I am looking to compute the following probability for a fixed $u$: $$ \Pr\{\exists i\in[n_1]: |x_i - u| < |y_j - u|, \forall j\in[n_2] \} $$ where the probability is to average over the randomness of the samples (which are drawn from the known Guassian distributions). I believe someone must have analyzed this before since I am essentially looking for the probability that the nearest neighbor to point $u$ is from a given class (when I know the generative models). Then, once this probability is computed for a fixed point $u$, the other part of the question is to how to use this result to obtain the desired probability when $u$ is drawn uniformly at random over the interval $[a,b]$. Any help is appreciated.
So following up on this, if I use the nice hint by @joriki for the dense case, I get that the desired probability is approximated by the integral $$ \frac{1}{b-a}\int_a^b \frac{n_1 f_1(u)}{n_1 f_1(u) + n_2 f_2(u)} du $$ Is there a way to approximate this integral which involves ratios of Gaussian functions along with the order of the approximation? The one idea that comes to mind is to condition on the values of $u$ for which $g(u) := f_2(u)/f_1(u) \leq \eta$, for some $\eta$ and to divide the integral into regions.