Let a general net of granularity δ be a partitioning of the region S into cells such that each cell fits inside a circle of diameter δ. The cells could be rectangles, parallelograms, polygons, or any shape defined by a simple closed curve. (The word cell is a bit misleading; the various cells could be completely different in size and shape.)
The lower sum is the ssum, over all the cells, of the area of each cell times the minimum of f on that cell. Upper sums are defined similarly. An arbitrary sum selects a random value from each cell, computes f at that point, and multiplies by the area of the cell. Once again the lower sum ≤ the arbitrary sum ≤ the upper sum.
Fix a Riemann net over the shape S. This net has a finite number of vertical lines, each of a given length, and a finite number of horizontal lines, each of a given length. Add these up to obtain a total length t.
Since f is integrable it is bounded. Let l be a lower bound and let u be an upper bound.
Choose a general net of granularity ε, where ε is small (we'll quantify this later). Most of the cells in this general net are wholly contained in one of the rectangular regions in the Riemann net. consider one of these "internal" cells, having area j. When computing the lower sum, j is multiplied by the minimum of f on this cell. This minimum value is greater than or equal to the minimum value of f over the containing rectangle. Use the minimum value of the rectangle instead, and the lower sum on the general net might decrease. Still, it is no smaller than the lower sum on the Riemann net.
Next consider a cell that is not internal; it intersects one of the lines in the Riemann net. In the Riemann sum its area might have been multiplied by u. In the general sum its area could be multiplied by l. This is a worst case. The difference is u-l times the area of the cell. Take all these cells together and build an error term that is no worse than 2(u-l)tε. This term approaches 0 as ε approaches 0. Therefore general nets produce lower sums that are bounded above the lower sum on the given Riemann net, minus an error term that goes to 0. The same holds for upper sums.
Remember that the lower and upper sums on Riemann nets converge to the integral. As they converge, the general nets produce lower and upper sums that are not far away from the Riemann lower and upper sums. Therefore the general sums converge to the same integral. Almost any tiling of the region S is fair game.
As a corollary, we can use any coordinate system to integrate f on S. Rectangles can be slanted this way and that; as long as they shrink to zero the integral is the same.
General nets can be built in n dimensions. I stayed in the plane for convenience.