# Urgent Question: Supp of a set of stochastic variables!

1. Nov 29, 2006

### Math_Ninja25

1. The problem statement, all variables and given/known data

Hi

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

I know that according to the definition

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

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

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

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

where $$\lambda > 0$$

I need to determine supp $$P_{Y,Z}$$

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

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

2. Relevant equations

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

3. The attempt at a solution

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

Sincerely Yours
Math_Ninja