In topology, the standard definition of continuity works in the "backward" direction, since it puts a condition on the pre-images of a function rather than images:
$f$ is continuous if the pre-image under $f$ of every open set is open.
However, there is an alternative definition of continuity that works in the "forward" direction:
$f$ is continuous if $p\in\overline A\implies f(p)\in\overline{f(A)}$.
The standard definition of measurable functions is analogous to the first definition of continuity (just replace "open" with "measurable"). Can measurable functions be defined without reference to pre-images, by analogy with continuity?
EDIT: The original version of this question did not specify that I was looking for a definition without any use of pre-images. The proposed duplicate has an answer that considers the "forward" direction, but it still uses pre-images in part of its formulation. First I made a new post with a refined question, but was then kindly instructed to re-open this one.