  The paper presents a new formula for the fractional integration, which
generalizes the Riemann-Liouville and Hadamard fractional integrals into a
single form, which when a parameter fixed at different values, produces the
above integrals as special cases. Conditions are given for such a generalized
fractional integration operator to be bounded in an extended Lebesgue
measurable space. Semigroup property for the above operator is also proved.
Finally, we give a general definition of the Fractional derivatives.
