I have started doing Lebesgue integration and I just want to clarify one thing to start with.
Very loosely speaking, with Riemann integration we partition the domain into $n$ intervals, and then we calculate the area of the $n$ rectangles formed by a base of width $n$ and a height of where each rectangle 'hits' the function. Summing these rectangles gives us an approximation of the area of the graph under the function. Then as we let $n \to \infty$ this approximation increase in accuracy and equals the function when we take the limit.
Now with Lebesgue integration do we follow the same process of partitioning (the range this time) into $n$ intervals and then letting $n \to \infty$ giving us smaller and smaller intervals which implies the approximation to the function keeps increasing? Leaving aside the concept of having sets that are measurable on the domain...I am simply wondering is process of considering intervals of decreasing size the same as with Riemann integration?