What is the Difference Between Riemann Integral and Lebesgue Integral?

The Riemann Integral and Lebesgue Integral are two different ways of defining the integral of a function. The main differences between them are:

1. Subdivision: The Riemann Integral subdivides the domain of a function, while the Lebesgue Integral subdivides the range of the function.
2. Approximation: The Riemann Integral approximates the assigned function by piecewise constant functions in each sub-interval, while the Lebesgue Integral approximates the function using simple functions.
3. Measurement: The Riemann Integral measures the 'height' of the function above a given part of the domain, while the Lebesgue Integral measures the 'height' of the function for a given part of the range.
4. Integrability: The Riemann Integral considers the area under a curve as made out of vertical rectangles, while the Lebesgue Integral considers horizontal slabs that are not necessarily rectangles.

The Lebesgue Integral was developed to address the limitations of the Riemann Integral, particularly when dealing with unbounded functions and taking limits of sequences of functions. The Lebesgue Integral is more flexible and allows for a broader class of functions to be integrated, including those that do not have a Riemann Integral.

Comparative Table: Riemann Integral vs Lebesgue Integral

The main differences between Riemann Integral and Lebesgue Integral are the way they measure the area under a curve and the types of functions they can handle. Here's a comparison table highlighting the key differences:

In summary, the Riemann Integral focuses on the domain of a function and its subintervals, while the Lebesgue Integral is concerned with the range of a function and its measurable sets. The Lebesgue Integral is more general and can handle a wider range of functions, including discontinuous ones, whereas the Riemann Integral is limited to continuous functions with compact ranges.