Suppose I want to count the number of points of $\mathbb{Z}^2 \subset \mathbb{R}^2$ which have length at most $L$.
This is equivalent to counting the number of points of $\frac{1}{L} \mathbb{Z}^2$ contained in the unit ball.
I am reading a set of notes that states that the number of points of $\frac{1}{L} \mathbb{Z}^2$ contained in the unit ball is asymptotic to $L^2$ times the Lebesgue measure of the unit ball.
The notes state that the Lebesgue measure (of what?) can be defined as the limit as $L \to \infty$ of $$\frac{1}{L^2} \sum_{\alpha \in \mathbb{Z}^2} \delta_{1/L \alpha}$$ where $\delta_x$ denotes the point mass at $x$.
(1). What is the above limit the Lebesgue measure of?
(2). Why is the number of points of $\frac{1}{L} \mathbb{Z}^2$ contained in the unit ball asymptotic to $L^2$ times the Lebesgue measure of the unit ball?
PS: I don't know what field of mathematics this type of question belongs to, any reference recommendation would be appreciated!
This is called the Gauss circle problem. Intuitively, you are shrinking the integer lattice by a factor $L$ in all directions. Each cell represents an area of $\frac 1{L^2}$. The limit counts these cells. The number of cells increases as $L^2$. The number of questionable cells (those near the perimeter so they might be in or out) only increases as $L$, so the error decreases as $\frac 1L$