Question

Prove that if two Riemann integrable functions [a, b] → R are

equal almost everywhere, then their integrals are equal.

Answer 1

To prove that two Riemann integrable functions equal almost everywhere have equal integrals, we need to consider the definition of 'equal almost everywhere' and recognize that any differences occur on a set with zero measure, hence not affecting the integral's value.

Step-by-step explanation:

To prove that if two Riemann integrable functions on the interval [a, b] are equal almost everywhere, their integrals are also equal, we follow these steps:

1. Define what it means for functions to be equal almost everywhere.
2. Understand the concept of a Riemann integral and how it is evaluated using limits of integration and summing the areas under the curve.
3. Apply the definition of functions being equal almost everywhere to conclude that differences occur only on a set of points with zero measure, which does not affect the value of the integral.

Since the set where the functions differ has measure zero, the difference in area under the curves is also zero, leading to the conclusion that the integrals of the functions are equal.