Pointwise convergence of functions

Let and . We say that pointwise if for all :
.

It does not preserve nice properties

Example 1

All CTS, but not CTS
for , otherwise
for , but

Example 2

All integrable, but not integrable.
Take for , otherwise.
Enumerate rational numbers in as and set for , otherwise set .

Example 3

All integrable, integrable, but
Take and for , otherwise.