We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integrals such as the Bounded Convergence Theorem, the Dominated Convergence Theorem, and the Second Mean Value Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form a Euclidian computable sequence of reals which converges effectively to zero. This property of convergence is the effectivization of the Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version (...) of Dirichlet's test. (shrink)

The subject of the present article is the following fact. Consider an effective uniform space. A generally constructed metric from the uniformity has the property that a sequence from the space effectively converges with respect to the uniform topology if and only if it does with respect to the induced metric. This can be shown without assuming the computability of the metric.