Neyman–Pearson Lemma

I am trying to wrap my head around the Neyman–Pearson lemma for simple vs simple hypothesis that is

$$H_o: theta_0 hspace{19mm} H_a: theta_1$$

with the respective pdfs $f_1$ and $f_0$. I am trying to understand when randomization of the hypothesis test occurs. We reject the null with probability $gamma$ for the following set ${yin Y: f_1(y)=kf_0(y)}$ where $Y$ is the range of our rv $X$ and $k$ is a constant chosen to get the appropriate size of the test.

When is randomization needed? I have seen randomization applied to both continuous and discrete random variables for composite/simple hypothesis ($H_o: text{Uniform}(0,10), H_a: text{Uniform}(2,12)$ randomization on $Xin(2,10)$). In a simple vs simple hypothesis testing for continuous random variables is randomization not required or is it a case by case basis and RV type does not tell us anything about randomization?

I was thinking about a continuous random variable $X$ then if I consider the set $A_k={yin Y: f_1(y)=kf_0(y)}$ and if $P(A_k)>0$ for all $kin R$. This would contradict $P$ being a probability measure. .Is this example correct or even enough to justify that for continuous RV the set $A_k$ must be a null set?