We know that the relation between uniform convergence and integration in case of interchangeability. I.e. Suppose and are integrable and fn--->f uniformly on [,] . Then
lim→∞(a to b)∫fn(x)dx=(a to b)∫f(x)dx. But I want some weaker condition than uniform convergency. I have seen it a question as follows. If fn are sequence of non- negative continuous function on [a,b]. which of the following conditions are sufficient for interchangeability of limit and integral as above. 1) fn(x) monotonically increases to f(x) for all x in [a,b]. 2) fn(x)<=f(x) for all x in [a,b]. 3) f is continuous. I think the first two option are like Monotone convergence theorem an Dominated convergence theorem in Measure Theory, but I can not simplify it to Reimann Integration case. Please someone help me to do.
