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Math_Ninja25

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## Homework Statement

Hi

I have been working on understanding concept fra Measure Theory known as support or supp

I know that according to the definition

if [tex](\mathcal{X},\mathcal{T})[/tex] is a topological space and [tex](\mathcal{X},\mathcal{T}, \mu)[/tex] such that the sigma Algebra A contain all open sets [tex]U \in \mathcal{T}[/tex]. Then the support on a measure [tex]\mu[/tex] is defined as the set of all points [tex]x \in \mathcal{X} [/tex] for which every open neighbourhood of x has a positve measure

[tex]\mathrm{supp}(\mu) = \{x \in X| \forall z \in Z \in N_{x} \in \mathcal{T}, \mu(N_{x}) > 0 \} [/tex]

Can this definition be used on a discrete stohastic vector [tex](Y,Z)[/tex] with the distributionfunction [tex]p_{Y,Z}[/tex]

[tex]P(Y = y, Z = z) = p_{Y,Z}(Y,Z) = \left\{ \begin{array}{ll}\frac{1}{2} \cdot e^{-\lambda} \frac{\lambda^z}{z!} & \mbox{where \ \mathrm{y $\in \{-1,0,1\}$} \mathrm{and} \mathrm{z $\in \{0,1,\ldots\}$}} \\ 0 & \mbox{\mathrm{other.}}\end{array} }\right[/tex]

where [tex]\lambda > 0[/tex]

I need to determine supp [tex] P_{Y,Z}[/tex]

Then by the definition of supp then the non-negative mesure on a meassurable space (Y,Z), is the function

[tex]P: \rightarrow ]-1,1[ \ \mathrm{x} \ ]0,\infty [[/tex]

## Homework Equations

The supp is is it then the set of all subsets which the measure operates on?

## The Attempt at a Solution

Therefore the support [tex]supp \ p_{(Y,Z)} = supp(P) = \{y \in Y| \forall z \in Z , P(Y,Z) > 0\}[/tex]

Sincerely Yours

Math_Ninja