dfa;lj

Measure space (Probability Measure - Dirac measure)

Let \(\displaystyle (\Omega,\mathfrak{F})\) be a measure space. Let \(\displaystyle M_1,...,M_n\) be measures on \(\displaystyle (\Omega,\mathfrak{F})\). Let \(\displaystyle a_1>0,...,a_n>0\). Define
\(\displaystyle M=a_1M_1+...+a_nM_n\)

For any \(\displaystyle A\in \mathfrak{F}\),
\(\displaystyle M(A)=\sum a_iM_i(A)\) (don't know how to do latex but above the sigma is 'n' and below the sigma is 'i=1'

Prove that M is a measure on \(\displaystyle (\Omega,\mathfrak{F})\)

Conclude from here that, for any points, \(\displaystyle x_1,...,x_n\in \Omega\) and any \(\displaystyle a_1>0,...,a_n>0\)

\(\displaystyle P=a_1, \delta_x_1+...a_1, \delta_x_1\)
is a measure on \(\displaystyle (\Omega,\mathfrak{F})\). Here \(\displaystyle \delta_x_1\) is the Dirac measure at \(\displaystyle x_1\). When is P a probability measure?
Hint: You may use without proof that each \(\displaystyle \delta_x_1\) is indeed a measure.

so yeah any help, thanks a lot.