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I recently asked this question about whether measurable functions can be alternatively defined in the "forward" direction, by analogy with an alternative definition of continuity (see below, or see linked post). The question was marked as a duplicate of this post, which gives an alternative definition of measurable functions.

However, I do not find the answer in the duplicate post acceptable, because it still references pre-images. So let me refine my previous question here:

Can measurable functions be defined without reference to pre-images, by analogy with continuity?

The analogous continuity definition, referenced in my previous post, is: $f:X\to Y$ is continuous iff for all $p\in X,A\subseteq X:p\in\overline A\implies f(p)\in\overline{f(A)}.$ This definition only involves the "forward" direction of $f$; meanwhile, the alternative definition of measurable functions in the linked question defines a pushforward of a filter $F$ in terms of pre-images. To me this seems to defeat the purpose of formulating a "forward" definition.

EDIT: Please refer back to the original question. I edited that post to ask the same question as in this one, and it was eventually reopened.

WillG
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    Where did you get the idea you can abandon a post to reask an edited version of it?? Your question https://math.stackexchange.com/questions/4202466/forward-characterization-of-measurable-functions was closed. Rather than posting your question again, with more details, you should have edited the linked question, for reopening. – amWhy Jul 19 '21 at 19:06
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    @amWhy Where did I get the idea? Here is the text displayed in the original question: "Your post has been associated with a similar question. If this question doesn’t resolve your question, ask a new one." – WillG Jul 19 '21 at 19:11
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    If the preferred practice is to reopen the old one, I'm happy to oblige. But please note that I'm an adult, not a child—I do not appreciate the scolding tone. – WillG Jul 19 '21 at 19:12
  • Yes, I confirmed, the correct practice is edit your first question to clarify what you meant the first time around, but failed to ask then. – amWhy Jul 19 '21 at 19:23
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    Only one more reopen vote needed. Thanks for editing there. I voted to reopen the former post of yours, after you edited. Thanks! – amWhy Jul 19 '21 at 20:21

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