Difference Between Riemann Integral and Lebesgue Integral

Riemann Integral vs Lebesgue Integral

Integration is a main topic in calculus. In a broder sense, integration can be seen as the reverse process of differentiation. When modeling real-world problems, it is easy to write expressions involving derivatives. In such a situation, the integration operation is required to find the function, which gave the particular derivative.

From another angle, integration is a process, which sums up the product of a function ƒ(x) and δx, where δx tends to be a certain limit. This is why, we use the integration symbol as ∫. The symbol ∫ is in fact, what we obtain by stretching the letter s to refer to sum.

Riemann Integral

Consider a function y=ƒ(x). The integral of y between a and b, where a and b belong to a set x, is written as _b∫^aƒ(x) dx = [F(x)]_a→b = F(b) – F(a). This is called a definite integral of the single valued and continuous function y=ƒ(x) between a and b. This gives the area under the curve between a and b. This is also called Riemann integral. Riemann integral was created by Bernhard Riemann. Riemann integral of a continuous function is based on the Jordan measure, therefore, it is also defined as the limit of the Riemann sums of the function. For a real valued function defined on a closed interval, the Riemann integral of the function with respect to a partition x₁, x₂, …, x_n defined on the interval [a,b] and t₁, t₂, …, t_n, where x_i ≤ t_i ≤ x_i+1 for each i ε {1, 2, …, n}, Riemann sum is defined as Σ_{i=o to n-1} ƒ(t_i)(x_i+1 – x_i).

Lebesgue Integral

Lebesgue is another type of integral, which covers a wide variety of cases than Riemann integral does. The lebesgue integral was introduced by Henri Lebesgue in 1902. Legesgue integration can be considered as a generalization of the Riemann integration.

Why do we need to study another integral?

Let us consider the characteristic function ƒ_{A (x) =} {_{0 if, x not ε A}^{1 if, x ε A} on a set A. Then finite linear combination of characteristic functions, which is defined as F(x) = Σ a_iƒE_i(x) is called the simple function if E_i is measurable for each i. The Lebesgue integral of F(x) over E is denoted by _E∫ ƒ(x)dx. The function F(x) is not Riemann integrable. Therefore Lebesgue integral is rephrase Riemann integral, which has some restrictions on the functions to be integrated.