Question: Assume that $f_X(x)$ is the probability density function of a random variable $X$ such that $X\in[a,b]$ for $b\gt a$.
Does there exist another random variable $Y\in[a,b]$ having $\dfrac{e^{2y}}{\mathbb{E}[e^{2X}]} f_X(y)$ as its probability density function?
Note: There reason I'm asking this question is that I want to treat $\dfrac{e^{2y}}{\mathbb{E}[e^{2X}]} f_X(y)$ as the pdf of a new random variable like $Y$ and somehow change the measure I'm working with (cause dealing with that $Y$ is much more simple for me in some situation). I believe the "Radon–Nikodym derivative (density)" should be related to my question. However, I'm not very familiar with the probability measures.
My try:
I know that the integral of the probability density function over the whole support of the variable should be equal to $1$. So I tried to prove that:
$$ \int_{a}^{b} \dfrac{e^{2y}}{\mathbb{E}[e^{2X}]} f_X(y) dy =1 $$
However, I do not know how to deal with the fraction $\dfrac{e^{2y}}{\mathbb{E}[e^{2X}]}$ in the integral. We know that $\int_a^b f_X(x)dx=1$. I cannot see what happens when the fraction is multiplied, and how it affects the whole integral.
The result follows from $$\int_a^b e^{2y} f_X(y) dy = \mathbb E[e^{2X}].$$ Also note that the function is nonnegative.