Question

If the following statement is true prove it, and if it is false then provide a counterexample: If two functions f,g : [a,b] → R are equal almost everywhere, and if f is Riemann integrable, then g must be too.

Answer 1

The statement that if functions f and g are equal almost everywhere and f is Riemann integrable, then g is also Riemann integrable, is true because g shares the same properties of boundedness and a set of discontinuities of measure zero as f.

Step-by-step explanation:

In mathematics, if two functions f and g are said to be equal almost everywhere on an interval [a,b], this means that they differ only on a set of measure zero; in other words, they are equal for all x in [a,b], except possibly at some isolated points which do not affect the value of integrals. Considering Riemann integrability, a function is Riemann integrable if it is bounded and its set of discontinuities has measure zero.

If the function f is Riemann integrable on [a,b] then it meets these criteria. Since g is equal to f almost everywhere, it differs from f only on a set of measure zero, which implies that the set of discontinuities of g also has measure zero. Moreover, since f is bounded and equal to g almost everywhere, g must also be bounded. Therefore, g is Riemann integrable by meeting the same conditions as f. Hence, the statement is true.