If we define two sets as(adsbygoogle = window.adsbygoogle || []).push({});

[tex] A = \left\{ (x,y)\in \mathbb{R} \vert x+y<a \right\} [/tex]

[tex] B = \left\{ (x,y)\in \mathbb{R} \vert x-y<a \right\} [/tex]

and then define the system of sets,

[tex] \mathbb{D} = \left\{A\vert a\in \mathbb{R}\right\}\cup\left\{B\vert a\in \mathbb{R}\right\} [/tex]

and then let [itex] \sigma(\mathbb{D}) [/itex] be the [itex] \sigma [/itex]-algebra generated by the system of set [itex] \mathbb{D} [/itex]. To show that the sets

[tex] C = \left\{ (x,y)\in \mathbb{R} \vert x+y\leq a \right\} [/tex]

[tex] D = \left\{ (x,y)\in \mathbb{R} \vert x-y\leq a \right\} [/tex]

are elements in [itex] \sigma(\mathbb{D}) [/itex] for every [itex] a\in \mathbb{R} [/itex].

Some questions arises. Can the sets A and B be considered aselementsin [itex] \mathbb{D} [/itex] and hence in [itex] \sigma(\mathbb{D}) [/itex] too, or are they justsubsetsof [itex] \mathbb{D} [/itex]?

Now I want to express C as a union of A and the set [itex] S = \left\{ (x,y)\in \mathbb{R} \vert x+y=a \right\} [/itex], and similar for D. But then I have to express the new set in some way to show that it is contained in [itex] \sigma(\mathbb{D})[/itex]. (Maybe there are easier approaches?). I was considering to write the new set S as

[tex] S = \bigcap_{n=1}^{\infty}\left\{ (x,y)\in \mathbb{R} \vert x+y<a+\frac{1}{n} \right\}\cap \left\{ (x,y)\in \mathbb{R} \vert x+y\geq a \right\} [/tex]

Where the last set is the complement of A. But will this set even converge towards S, or will it just be empty?

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# Homework Help: Sigma-Algebra problem

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