The following definition is a variant of what is called the

The following definition is a variant of what is called the Darboux integral and could easily seen to be equivalent to the Riemann integral of elementary calculus fame: Let f: R rightarrow R be a function with compact support and define I^D^+(f):= inf {integral t: t gt f and t step function}_D^-(f):= sup{integral t: t lt f and t step function} If D_+(f) = D_(f), wc call f Darboux integrable (which is equivalent to Riemann integrable) and the common value the Darboux integral of f. Prove that if f is Darboux integrable, then f is Lebesgue integrable, and D^+(f) = D^-(f) = integral f.

Solution

Riemann Integral states that the area under the curve using definite integral between two given limits is the sum of all the areas with maximum of all lower bounds = sum of all the areas with minimum of all upper bounds. Here in the Darboux for f: R to R , it is defined as

D⁺(f) = Infinimum ( t : t f and t is step function )

D^-(f) = Sup ( t : t f and t is step function)

As D⁺(f) =D^-(f), it is a Darboux integrable and D⁺(f) =D^-(f) = f, it is Lebesque integrable.