vector spaces: verify axioms for 1, random variables and 2, power set of a set over {0,1}

Question

Verify the axioms of a vector space for:

1. The set of all real-valued random variables on a fixed sample space over $R$
   
2. The power set of a set $\Omega$ forms a vector spaces over $F=${$0,1$}
   
   under the usual operations.
   

What I know are the axioms of a vector space, $V$ over field, $F$. These are:

1. $V$ is a commutative group (abelian group)
   
2. $\centerdot$ is a map from $F \times V$ to $V$
   
3. $\alpha\centerdot(\beta\centerdot x)=(\alpha\beta)\centerdot x$
   
4. $1\centerdot x=x$
   
5. $(\alpha+\beta)\centerdot x=(\alpha\centerdot x)+(\beta\centerdot x)$
   
6. $\alpha\centerdot(x+y)=(\alpha\centerdot x)+(\alpha\centerdot y)$
   
   

I have just started learning vector spaces and am unable to extend my concept properly and am not sure how to verify the axioms for these two examples. Even a verification for at least the first property for each would be enough to make my concept clearer. It is easy to verify for a group where the elements and the two standard operations on them have been defined, but I am not sure how to do that here. Please, help!

Answer 1

For any subset $A$ of $\Omega$ define $0\cdot A=\emptyset,\,1\cdot A=A$. For any subsets $A,\,B$ define $A+B$ as the disjoint union of $A,\,B$ (this requires the convention $1+1=0$). The singletons form a basis of the space, with $A$ the sum of the singletons that are subsets of it.