Für jede natürliche Zahl $n \in \mathbb{N}$ sei $A_n$ die von der Mengenfamilie $\{\{1\}, \{2\}, \dots , \{n\}\}$ erzeugte sigma-algebra auf $\mathbb{N}$. Zeigen Sie, daß $A_n$ neben $B = \emptyset$ und $B = \mathbb{N}$ aus allen Mengen $A \subset \mathbb{N}$ besteht, welche entweder $A \subset \{1, \dots, n\}$ oder $m \in B$ für alle $m \geq n +1$ erfüllen

This is what I have come up with:

For every natural number $n \in \mathbb{N}$, let $A_n$ be the family of sets $\{\{1\}, \{2\}, \dots , \{n\}\}$ which generates a sigma-algebra on $\mathbb{N}$. Show that $A_n$ [...]


1 Answer 1


For every natural number $n \in \mathbb N$, let $A_n$ be the $\sigma$-algebra on $\Bbb N$ generated by $\{ \{1\},\cdots,\{n\} \}$. Show that $A_n$ consists, besides $B = \varnothing$ and $B = \mathbb N$, of all subsets $B$ which satisfy $B \subseteq \{1,\cdots,n\}$ or $m \in B$ for all $m \ge n+1$.

Note 1 : There were obvious typos in the question, I also corrected them.

Note 2 : The cases $B = \varnothing$ and $B = \mathbb N$ do not need to be excluded... this exercise looks like it's been written by a very strange mathematician.

Hope that helps,

  • $\begingroup$ The $B = \emptyset, \ B = \mathbb{N}$ was actually $A = \emptyset, \ A = \mathbb{N}$ and $A_n$ was this "fancy" sigma-algebra $A$, but I do not know how to type that in Latex. Otherwise I just ctrl + c, ctrl + v. I'm studying in Munich so there are maybe local words... $\endgroup$
    – Olba12
    Nov 2, 2016 at 0:11
  • $\begingroup$ @Olba12: Do you mean $\mathcal{A}$, $\mathscr{A}$, or $\mathfrak{A}$? (They are \mathcal{A}, \mathscr{A}, and \mathfrak{A}, respectively.) $\endgroup$ Nov 2, 2016 at 0:14
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    $\begingroup$ I mean $\mathcal{A}$, thanks! @BrianM.Scott $\endgroup$
    – Olba12
    Nov 2, 2016 at 0:15
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    $\begingroup$ @Olba12 : You might want to have a look at this : detexify.kirelabs.org/classify.html There's even an app for your phone (called Detexify) so you can look for the LaTeX code of some fancy symbols! $\endgroup$ Nov 2, 2016 at 0:24
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    $\begingroup$ @Olba : There are no local words in that quote of yours. However, in the same way that mathematicians use words like 'henceforth' which are barely (if not never) used by most people, the mathematical language is full of it. Analogously, some German words show up more often than other in mathematics compared to everyday life German. For instance, even though "erzeugen" is not a very fancy German word (I am thinking of fancy words like dementsprechend or daraufhin), it is very common in mathematics (since it means "to generate", and many things generate many other things in mathematics). $\endgroup$ Nov 2, 2016 at 0:27

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