# Convex combination of measures define a new probability measure, $a\mathbb{P}+b\mathbb{Q}$

Show that if $$\mathbb{P}$$ and $$\mathbb{Q}$$ are two probability measures defined on the same (countable) sample space, then $$a\mathbb{P} + b\mathbb{Q}$$ is also a probability measure for any two nonnegative numbers $$a$$ and $$b$$ satisfying $$a + b = 1$$.

I understand what I have to do but mathematically I don't know how to prove this

You need to show that $$\mathbb{P}^*=a\mathbb{P}+b\mathbb{Q}$$ satisfies the probability measure definition. Let $$(\Omega,\mathcal{F})$$ be a measurable space, $$\mathbb{P}^*$$ is a measure onto it if it satisfies the following conditions,
1. $$\mathbb{P}^*:\mathcal{F}\to[0,1]$$
2. $$\mathbb{P}^*$$ is $$\sigma$$-aditive: If $$(A_k)_{k\geq 1}$$ is a sequence of disjoint sets in $$\mathcal{F}$$ then $$\mathbb{P}^*(\bigcup_{k\geq1}A_k)=\sum_{k=1}^\infty\mathbb{P}^*(A_k)$$
3. $$\mathbb{P}^*(\Omega)=1$$
• @jamessmithh2002 A tip: We know that the above hold for $\mathbb{P}$ and $\mathbb{Q}$ isolated, try working with that a little bit. Commented Feb 16, 2022 at 14:29