Question

A3. Let (Ω, F, P) be a probability space and let H ∈ F with P(H) > 0. For any arbitrary

A ∈ F, let

PH(A) =

P(A ∩ H)

P(H)

Show that (Ω.F, PH) is a probability space.

Answer 1

We need to show that $PH$ is a probability on the sigma-algebra F. Indeed:

$PH\left( A \right) =\frac{P\left( A\cap H \right)}{P\left( H \right)}\geqslant 0\\PH\left( \varOmega \right) =\frac{P\left( \varOmega \cap H \right)}{P\left( H \right)}=\frac{P\left( H \right)}{P\left( H \right)}=1\\A_i\cap A_j=\emptyset :PH\left( \sum{A_i} \right) =\frac{P\left( \sum{A_i}\cap H \right)}{P\left( H \right)}=\frac{P\left( \sum{\left( A_i\cap H \right)} \right)}{P\left( H \right)}=\\=\frac{\sum{P\left( A_i\cap H \right)}}{P\left( H \right)}=\sum{\frac{P\left( A_i\cap H \right)}{P\left( H \right)}}=\sum{PH\left( A_i \right)}$