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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1$\begingroup$ Hint: see what (visually) happens when we deal with $\mathbb{R}^2$. $\endgroup$– KolminCommented Feb 6, 2015 at 11:08
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2$\begingroup$ 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"? $\endgroup$– MickGCommented Feb 6, 2015 at 11:37
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4$\begingroup$ 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. ;) $\endgroup$– KolminCommented Feb 6, 2015 at 18:03
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1$\begingroup$ 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. $\endgroup$– MickGCommented Feb 6, 2015 at 19:01
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2$\begingroup$ @MrArsGravis . In geometry a cylinder is $D\times \mathbb R$ where $D$ is a disc. In measure theory the disc can be anthing. That's it. In maths it is futile to get headaches about nomenclature. $\endgroup$– Kurt G.Commented Jan 25, 2023 at 13:28
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1 Answer
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What is the origin of this name?
Here is my guess (this is essentially what I was told in one of my lectures and what Kurt G. wrote in the comments):
If $|J|=1$, i.e. $J=\{i\}$ for some $i\in I$, then the cylinder sets are simply the preimages of measurable subsets of $E$ w.r.t. the projection$$\pi_i:E^I\to E.$$Now why are the preimages of a projection called cylinder sets? Of course this is motivated by a very specific example:
Let $E^3$ be a $3$-dim. euclidean space and $E^2$ a $2$-dim. subspace. Now we can consider the obvious projection $E^3\to E^2$ (i.e. $x\in E^3$ is mapped to the intersection of $E^2$ and the line through $x$ perpendicular to $E^2$) and in this case it should be clear why the preimages are referred to as cylinders: At least the preimages of circles in $E^2$ are (infinite) cylinders in the usual sense.
In case that you complain that $E^3$ is not a product space: Pick any 1-dim. subspace $E^1$ orthogonal to $E^2$ and consider the obvious bijection $E^1\times E^2\to E^3$.