family of functions/sequences taken over reals instead of naturals

Question

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as follows:

If $f:U \rightarrow \mathbb{R}$ is locally integrable, define its mollification $$f^{\epsilon} := \eta_{\epsilon}*f \text{ in } U_{\epsilon}$$

That is, $f^{\epsilon}(x) = \int_{U} \eta_{\epsilon}(x-y)f(y)dy = \int_{B(0,\epsilon)}\eta_{\epsilon}(y)f(x-y)dy$

for $x \in U_{\epsilon}$

are defined it is given that it has properties such as $f^{\epsilon} \rightarrow f \text{ a.e as } \epsilon \rightarrow 0$ and $f^{\epsilon} \rightarrow f$ uniformly on compact sets if $f$ is continuous on the domain. Are these definitions of point-wise convergence and uniform convergence the same as for the countable family of functions usually given as $(f_{n})_{n}$ with $n$ taken from the natural numbers? What about sequences $(x_{n})_{n}$ taken over $\mathbb{R}$ instead of $\mathbb{N}$?

By the way these definitions and properties of mollifiers are from the book 'partial differential equations' by Lawrence C.Evans.

Answer 1

To define convergence, we specify traps and tails. The definition always* says the same thing: convergence happens if every trap catches [i.e., contains] some tail. Traps and tails can be indexed by however you want to index them; some examples are given below.

Traps (for convergence to $f$):

• Uniform convergence: $T_\epsilon = \{g : |g(x)-f(x)|<\epsilon \text{ for all $x$}\}$
• Pointwise convergence: $T_{x,\epsilon} = \{g : |g(x)-f(x)|<\epsilon\}$
• Convergence in $L^p$ norm: $T_\epsilon = \{g : \|g -f \|_{L^p}<\epsilon \}$
• Convergence in measure: $T_{\epsilon,\delta} = \{g : \mu(\{x : |g (x)-f(x) |\ge \delta\}) <\epsilon \}$

Tails

• for a sequence: $t_N = \{f_n : n\ge N\}$
• for a family indexed by a positive number: $t_\delta = \{f^\epsilon : \epsilon <\delta \}$
• for a net, $t_\alpha = \{f_\beta : \beta\succeq \alpha\}$

To directly address your question:

Are these definitions of point-wise convergence and uniform convergence the same as for the countable family of functions?

The traps are the same, the tails are different (see above).

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(*) A notable exception to the above is convergence almost everywhere, which does not fit into any topology. You are unlikely to see "$f^\epsilon \to f$ almost everywhere as $\epsilon\to 0$"; although this convergence could be defined for uncountable families, it is not natural for them.