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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

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1 Answer 1

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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. $\mathbb{P}^*(\Omega)=1$
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    $\begingroup$ 3 is easiest and once you understand that, 1 and 2 are also easy $\endgroup$
    – Henry
    Commented Feb 16, 2022 at 14:04
  • $\begingroup$ im struggling to understand why 3 holds $\endgroup$
    – jamessmithh2002
    Commented Feb 16, 2022 at 14:14
  • $\begingroup$ @jamessmithh2002 A tip: We know that the above hold for $\mathbb{P}$ and $\mathbb{Q}$ isolated, try working with that a little bit. $\endgroup$
    – BelwarDissengulp
    Commented Feb 16, 2022 at 14:29

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