Let X be any set and A = P(X) be the power set of X. Let x1,...,xn be distinct points in X and let α1, . . . , αn be positive real numbers. Show the measure on A.
I'm uncertain if μ = α1δx1 + α2δx2 + · · · + αnδxn defines a measure on A and would really appreciate some insight into why this may be.
If $x_i\in X$ and $\alpha_i\in[0,\infty)$ for $i=1,\dots,n$ then $\sum_{i=1}^n\alpha_i\delta_{x_i}$ denotes a measure on $\sigma$-algebra $\wp(X)$.
The measure is prescribed by: $$B\mapsto\sum_{i=1}^n\alpha_i\delta_{x_i}(B)=\sum_{i=1}^n\alpha_i\mathsf1_B(x_i)$$