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Let $X$ and $Y$ be two non-empty sets and $f:X\to Y$ a function. Furthermore, let $\tau$ be a topology and $\Sigma$ a $\sigma$-algebra, respectively, on $X$. It is easy to see that \begin{align*} f(\tau)\equiv&\;\{E\subseteq Y\,|\,f^{-1}(E)\in\tau\} \end{align*} is a topology on $Y$, while \begin{align*} f(\Sigma)\equiv&\;\{E\subseteq Y\,|\,f^{-1}(E)\in\Sigma\} \end{align*} is a $\sigma$-algebra on $Y$.

While there is a well-established terminology for $f(\tau)$: it is known as the final topology, it seems to be the case that there is no similarly widespread name for $f(\Sigma)$.

I tried searching

  • “final $\sigma$-algebra/$\sigma$-field” by analogy to the concept of the final topology; and

  • “pushforward $\sigma$-algebra/$\sigma$-field” by analogy to the concept of the pushforward measure,

but these do not seem to be standard expressions.

Can anyone suggest a terminology for $f(\Sigma)$ that is widely recognized?

triple_sec
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1 Answers1

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I think the induced $\sigma$-algebra by $f$ (and $\Sigma$) is a term that's used, but I couldn't find a source, and the "bible" (i.e. Fremlin's five (!) "Measure Theory" books), defines but doesn’t name it, at least not where I found it.

Henno Brandsma
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  • Yes, I can see it in Volume 1, 111X(c). In your opinion, would it be weird to call it “pushforward $\sigma$-algebra,” “image $\sigma$-algebra,” or “final $\sigma$-algebra”? – triple_sec Oct 19 '18 at 23:56
  • @triple_sec I found it in Fremlin too, just the fact that is is a $\sigma$-algebra, but not a name for it. I still prefer induced, though “final” or “image” is possible too. With filters it’s called the push-foward IIRC. – Henno Brandsma Oct 20 '18 at 05:29