Integral of Fine Computable functions and Walsh Fourier series

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integral such as Bounded Convergence Theorem and Dominated Convergence Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Finecomputable function form an E-computable sequence of reals and converge effectively to zero. The latter fact is the effectivization of Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version of Dirichlet’s test.