I am looking for an "easy" reference on the convergence of measures applied to the following:
I have a probability space $(\Omega_n, \mathcal{F}_n, P_n)$ where $\Omega_n$ is countably finite of size $n^2$ and $P_n$ is the counting measure.
I define a random variable on this probability space $X_n:\Omega_n\rightarrow \{0,1\}$
Assume that $\sum_{\omega\in \Omega_{n}}\mathbb{1}\{X_n(\omega)=1\}\leq n$.
Notice: $$ P_n(\{\omega\in \Omega_n \text{ s.t. } X_n(\omega)=1\})\equiv \frac{\sum_{\omega\in \Omega_{n}}\mathbb{1}\{X_n(\omega)=1\}}{n^2}\leq \frac{1}{n} $$
I am looking for a mathematical framework/result/notion (if it makes any sense) stating that the measure
$$ \frac{\sum_{\omega\in \Omega_{n}}\mathbb{1}\{X_n(\omega)=1\}}{n^2} $$ (or maybe a rescaled version?) "converges" to a Lebesgue integral as $n\rightarrow \infty $.
I am very confused because $$ \lim_{n\rightarrow \infty} \frac{\sum_{\omega\in \Omega_{n}}\mathbb{1}\{X_n(\omega)=1\}}{n^2} =0 $$ which leads me to think that I have to rescale, but beyond that I don't see what I should do.
Could you guide me towards some "easy" reference? I have read about weak or vague convergence of measures but it looks too wide and I can't read the whole topic in order to answer my doubt.