Timeline for The image of a measurable set under a measurable function.
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 2, 2022 at 7:28 | vote | accept | Alex M. | ||
| Apr 25 at 20:59 | |||||
| Apr 1, 2022 at 20:47 | comment | added | Michael | I believe the above comment considers only Borel measurable functions $f:V_1\rightarrow V_2$ where $V_1, V_2$ are Polish spaces, so $V_1 \times V_2$ is a Polish space, and the "continuous function" is the projection map $h:V_1\times V_2\rightarrow V_2$ given by $h(v_1,v_2)=v_2$, which can project the graph set $\{(v,f(v)): v \in V_1\}\subseteq V_1 \times V_2$ to the set $f(V_1)$. | |
| Dec 3, 2013 at 7:51 | comment | added | Stefan Geschke | Kechris' Classical Descriptive Set Theory should contain all the relevant information. The key word here is "analytic set". A set is analytic if it is a continuous image of a Borel set in a Polish space. Since graphs of Borel measurable functions are Borel, the image of a Borel set under a Borel measurable function is analytic, being the projection of a Borel subset of the graph of the function. And analytic sets are Lebesgue measurable (see the wikipedia entry on analytic sets.) | |
| Sep 25, 2013 at 13:59 | comment | added | Dan Glasscock | Can you provide a reference? | |
| Jun 9, 2013 at 6:09 | history | answered | Stefan Geschke | CC BY-SA 3.0 |