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?
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.