Random Probability of Random Variable

Let be a Probability Space Let be a discreet set. Let be a Random Variable Let be given by:

i.e. the distribution of . Consider now the Random Variable given by:

often written as just . From the definition of , you might naively substitute:

This is wrong. We need to fully expand the definitions to understand why. Firstly:

Now we can properly see:

i.e. is randomly sent to a probability in the image of

Thinking about it further, if takes value with probability then will take the value with probability