Borel summation is a summation method for divergent series, It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. and The main purpose of Lebesgue integral is to provide an integral notion where limits of integrals hold under mild assumptions then both seeking for convergence or any function to be defined as well , Really i want to know if there is any connection between Borel summation and Lebesgue integral in the context when a function represented in a power series, i want to know for example :Could be this: $f$ is Lebesgue integral $\iff$ it is Borel-summable ?
Note: Here $f$ is smooth but non-analytic over $\mathbb{R+}$ and no explicit formula for the sequence $a_n$