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If $I$ is any set of indexes, we define $E^I=\{(x_i)_{i\in I}:x_i\in E\,\,\forall i\in I\}$, $E$ being any set. Subsets of $E^I$ of the form $C_J=\{x_i\in B_i\,\,\forall i\in J\}$, where $B_i\in\mathcal{A}\,\,\forall i\in J$, $\mathcal{A}$ is a $\sigma$-algebra on $E$ and $J\subseteq I$ is finite, are called "cylinder sets", and form a basis of the product $\sigma$-algebra $\mathcal{A}^{\otimes I}$. Why are these sets called "cylinder sets"? What is the origin of this name?

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    $\begingroup$ Hint: see what (visually) happens when we deal with $\mathbb{R}^2$. $\endgroup$
    – Kolmin
    Feb 6, 2015 at 11:08
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    $\begingroup$ The cylinder sets of $\mathbb{R}^2$ are products of borel sets of the real line. For example rectangles, i.e. products of intervals. In $\mathbb{R}^3$, the cylinder sets are parallelepipeds. So @Kolmin why not "rectangles", "generalized rectangles", "rectangle sets" or "parallelepiped sets"? $\endgroup$
    – MickG
    Feb 6, 2015 at 11:37
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    $\begingroup$ Don't shoot the messanger... I did not invent the terminology. :D Btw, behind jokes, the hint I gave you is exactly the one I found in a book to see why cylinder sets are called in such a way. Moreover, regarding your terminology, it sounds a bit problematic, because – take $\Re^2$ for example – cylinder sets are not bounded above and below, i.e. they are not actually rectangles. Indeed, they are cylinders. ;) $\endgroup$
    – Kolmin
    Feb 6, 2015 at 18:03
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    $\begingroup$ To me, a cylinder is a solid with circular basis extending in the third dimension for a possibly infinite height. This can be seen as a rotation solid, thus matching this definiton. Now obviously you can get cylinders as cylinder sets if you view $\mathbb{R}^3$ as the product of the plane and the line, as a circle is Borel in the plane and the third dimension is an interval. However, if $\mathbb{R}^3$ is the product of three lines, it is not straightforward to get a cylinder, since you have to first create the circle. $\endgroup$
    – MickG
    Feb 6, 2015 at 19:01
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    $\begingroup$ Indeed, even as plane times line, you have to select a basis of the planar Borels which may seem less "simple" than the rectangles. And a rectangle to me can have one or even two infinite sides. Now I've seen the disambiguation page on Wikipedia, but the link "Cylinder (algebra)" sends me to cartesian products. Perhaps there is another use of the word cylinder which I have never heard of. @What say you @Kolmin? From your comment it most definitely seems so. $\endgroup$
    – MickG
    Feb 6, 2015 at 19:01

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