Let f ∈ L p ( R ) for some 1 ≤ p < ∞....
Let for some . Given , we set .
Here I'm not sure if we can view as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when , we have pointwise a.e. and ?
I'm also not sure if it's possible to apply results such as Dominated convergence theorem and Fatou's lemma when the functions are indexed like this, even if we were to set and let tend to infinity.