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 realworld 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}∫^{a}ƒ(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 x_{1}, x_{2}, …, x_{n} defined on the interval [a,b] and t_{1}, t_{2}, …, 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 n1} 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_{i}ƒE_{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.
What is the difference between Riemann Integral and Lebesgue Integral? · The Lebesgue integral is a generalization form of Riemann integral. · The Lebesgue integral allows a countable infinity of discontinuities, while Riemann integral allows a finite number of discontinuities.

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