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## Main Question or Discussion Point

Hello

Normally in order to change the order of limit and integration in rimann integration, you need

But lets say that you are not able to prove uniform convergence, but only pointwise convergence. And lets say you are able to prove that the functions are also bounded. Could you then use something like the dominated convergence theorem for lebesgue measurable functions? The problem is offcourse that it holds for integrals defined another way.

But lets say you have a sequence of riemann integrable functions, that converge

Normally in order to change the order of limit and integration in rimann integration, you need

**uniform**convergence.But lets say that you are not able to prove uniform convergence, but only pointwise convergence. And lets say you are able to prove that the functions are also bounded. Could you then use something like the dominated convergence theorem for lebesgue measurable functions? The problem is offcourse that it holds for integrals defined another way.

But lets say you have a sequence of riemann integrable functions, that converge

**pointwise**to a function, and all the functions are bounded, can we then change the order of limit and integration?