On the first page of chapter $1$ in Bauer's book Measure and Integration Theory one can read:
The limits of such elementary considerations are already reached in defining the area of an open disk $K$, to which end one proceeds thus: a sequence of open $3\cdot2^{n-1}$-gons $E_n$ ($n\in\mathbb{N}$) is inscribed in $K$, with $E_1$ being an open equilateral triangle, and the vertices of $E_{n+1}$ being those of of $E_n$ together with the intersections of the circle with the radii perpendicular to the sides of $E_n$. Thus $E_{n+1}$ consists of $E_{n}$ together with its $3\cdot2^{n-1}$ edges and the open isoceles triangles which have these edges as hypothenuses and vertices on the circle. Since $K$ is the union of of all the $E_n$, it looks like a "mosaic of triangles", that is, like a union of disjoint open triangles and segments (namely, common sides of various triangles). The following broader formulation of (B) therefore leads to a definition of the area of the disk $K$:
(C) If $(A_n)$ is a sequene of pairwise disjoint sets, and $A_n$ has numerical measure $\alpha_n$ ($n\in\mathbb{N}$), then $\cup_{n=1}^{\infty}A_n$ has numerical measure $\sum_{n=1}^{\infty}\alpha_n$.
So far so good but then Bauer writes:
If we replace $K$ and every $E_n$ by its topological closure, this method would not lead to a plausible definition of the area of a closed disk $\bar{K}$, because $\bar{K}$ is not the union of the closures $\bar{E}_n$ of the above constructed polygons $E_n$.
Why is this so? In my mind I have the following picture:
