Probability measure, probability density function or probability event ? Are they different?

Question

My question is regarding the difference between probability measure and probability of event.

Recently I have read an information theory paper that considered a channel modeled by probability density function (pdf) $$\mathbb{P}(Y_1\mid X_1)$$ The exact words of the paper are:

A memoryless channel with joint transition probability distributions (or conditional channel distribution: $\mathbb{P}(Y\mid X)$

while the probability of decoding error is defined as

$$\mathbb{P}[\widehat{W} \not= W]$$

I am wondering is there any difference between the two definitions of probability. As you can see one used the paranthesis P() and the other definition used the bracket P[] notation.

Thanks looking forward for your answers.

Answer 1

There is often an abuse of notation,

One writes $P[X \in A]$ to mean $P(\{\omega: X(\omega) \in A\})$ In the same sense for $B \subset \Omega$ measurable set one writes $P(B)$ to mean the probability of the event $B$. The notation is quite similar and one might find them misplaced.

But as it is a notation issue, one must be prepared to find different conventions and to adapt and interpret the different representations.