Lebesgue Integral is Extension of Darboux Integral

Theorem
If $$f:[a,b] \to \mathbb{R}$$ is Riemann integrable, then it is measurable and

$$R \int_a^b f(x)dx = \int_a^b f$$

where $$R \int_a^b$$ is the Riemann integral and the $$\int_a^b \ $$ is the Lebesgue integral.

Proof
Since every step function is also a simple function, we have

$$L(P) \leq \sup_{\phi\leq f} \int_a^b \phi(x)dx \leq \inf_{\psi\geq f} \int \psi(x)dx \leq U(P)$$

where $$L(P) \ $$ and $$U(P) \ $$ are the lower sum and upper sum as defined in the definition of definite integral.

Since $$f \ $$ is Riemann integrable, the inequalities are all equalities and $$f \ $$ is measurable by basic properties of measurable functions.