I am trying to solve an assignment of measure theory of an institute in which I don't study as our professor doesn't give any assignment. I am struct on this question. Can someone please tell how to proceed.
Let $S= \{ s_1 , s_2 , \ldots , s_ k \}$ , where $k \le \infty $ .
Let $B = \mathcal{P}(S)$ be the power set of $S$, and $ p_1, p_2, \ldots, p_k $ be non negative numbers .
Let
$$
\lambda(A) = \sum_{I=1}^k p_i I_A ( s_i ) ,
$$
where $ I _A $ is the characteristic (indicator) function of $A$.
I have to prove that
(1) $( S, B, \lambda )$ is a measurable space.
(2) Topology on which it is defined / topology which it induces.
Please help
To answer (1) - and here I am assuming from your notation that $S$ is a finite or countable set - we need to show that $\lambda$ satisfies the definition of a measure. The only tricky part involves the passage to the limit (i.e. countable additivity for disjoint sets), which can be justified using Tonelli's theorem since everything is non-negative.
For (2), I am assuming you mean to find a topology on $S$ whose Borel $\sigma$-algebra equals $B$. But since $B$ is the power set, all events are measurable so the topology is discrete (i.e., the topology also equals the power set - so in this case, the topology equals the $\sigma$-algebra). As a side note, this is why it is called discrete probability.