The statement "Measurable functions are closed under addition and multiplication, but not composition." was generated by Wolfram Alpha from a source of mixed quality by turning all references (and hyperlinks) from the original text into a "related topics" section with no connection to the main text. I have two questions related to this:
- Is automatic removal of references from a mathematical text (as done by Wolfram Alpha) an "invalid" text transformation? For example, if I write a text, do I have to take care that it won't change its meaning dramatically if all references are "silently" removed? (Checking my last publication, I realized that some sentences in it would also "break" if the references would be "silently" removed.) Should Wolfram Research fix this "automatism"?
- Is there something wrong (or evil) with the "subsets of sets with measure zero" modification that make it especially "fragile" with respect to minor inaccuracies? Isn't such a "fragile" concept difficult to remember correctly, and hence would require a really good justification by a corresponding significant benefit with respect to more "canonical" concepts?
Context (original question)
In theory, I think the ideas behind Wolfram Alpha are very worthwhile. In practice, google seems to be good enough for me. But now I tried to use Wolfram Alpha for my math questions. It was surprisingly good at understanding my questions. Most answers were disappointingly "shallow", but not wrong. However, the answer to measurable function deeply worries me:
... When $X=\mathbb{R}$ with Lebesgue measure, or more generally any Borel measure, then all continuous functions are measurable. In fact, practically any function that can be described is measurable. Measurable functions are closed under addition and multiplication, but not composition.
Both the statement that the Lebesgue measure is a Borel measure and the statement that measurable functions are not closed under composition are seriously misleading for me. (I had to read the Wikipedia article to reassure me that my current understanding wasn't completely wrong). You may object that I quoted this out of context, but the removal of context done by Wolfram Alpha is exactly what worries me. The original article from MathWorld certainly wasn't great, but the removal of the hyperlinks and references by Wolfram Alpha really killed it.
However, this also made me think about the usefulness of the "subsets of sets with measure zero" modification to the Borel measurable sets. Is there any good practical reason that justifies the confusion created by the difference between "Lebesgue measurable" and "Borel measurable"? Are there any practically relevant functions that are "Lebesgue measurable" but not "Borel measurable"?