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- TL;DR Summary
- Is there a type of convergence defined for sequences of functions that have different domains?

Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h = 1/n## and F_n(x) has the domain ##\{h,2h,3h,..1\}## Intuitively, the sequence of functions ##F_n, n = 1,2,3,...## converges to the function ##F(x) = 1## defined on ##(0,1]##.

Which definitions handle the above type of convegence

Which definitions handle the above type of convegence

*without*assuming the definition of each function in the sequence is extended to have the same domain as the function that is the limit? ( For example, for convergence in ##L_2## we could stipulate that the domain of each ##F_n(x)## is extended to ##(0,1]## by defining ##F_n(x)## to be a step function over intervals ##(kh, (k+1)h)## Then the same notion of integration would apply to both ##F_n(x)## and ##F(x)##. Can we avoid making such stipulations?)