"A first Course in probability"-chaper4-continuous type random variable-the expectation of random variable function

In connection with the introduction of discrete random variables, after defining the expected e[x of the random variable x], we tend to focus on the random variable e[g (x) of the function of x in the actual problem, and continue to discuss the expectation of the discrete random variable function, we can easily make the following conjecture:

However, we should note that in discrete random variables, because this function relationship g (X) does not change the probability distribution after, so the function of the discrete random variable is expected to solve the formula in our view is obvious. But in this case, the functional relationship g (X) obviously destroys the density function of the original random variable, which means that the density function of the random variable y=g (x) is not f (x), so here we pass the simple substitution, which is obviously insufficient to reflect the correctness of this speculation, So we need a more forceful proof process.

We can first try to solve E[g (X) by a topic:

Obviously we see here that the density function for the random variable y has changed and echoed what we said above.

So, how do we prove the first thing we gave?

"A first Course in probability"-chaper4-continuous type random variable-the expectation of random variable function