I'm a beginner in statistics and in particular in R.V. I need a slow and detailed explanation of the facts below. For an event $A$, I understand that the indicator random variable $1_A$ is $1$ if $A$ occurs, and $0$ otherwise. Why then $P(A)=E[1_A]$ and why $$E_\theta[1_A]=E_\theta[E_\theta[1_A∣X]]\ ?$$ Anyway, what is $E_\theta$ here ?
EDIT :
We consider two definitions of the power function which equivalence is unclear to me.
(1) The power function of a test, denoted by $Q(\theta)$ is the probability of rejecting null hypothesis $H_0$ when $\theta\in \Theta$ is the true parameter value. The power function is then given by $$Q(\theta)=P_\theta(\cal R).$$
We have obviously here $Q(\theta_0)=\alpha$ and $Q(\theta_1)=1-\beta$ where $\alpha$ is the I type error and $\beta$ is the II type error.
(2) A function $\psi(.):\cal X^n\to [0,1]$ is called the critical function where $\psi(\mathbb{x})$ stands for the probability with which the null hypothesis $H_0$ is rejected when the data $\mathbb{X}=\mathbb{x}$ has been observed, $\mathbb{x}\in \cal X^n$.
Why now, we can rewrite the power function defined in (1) as follows
$$Q(\theta)=E_\theta\{\psi(\mathbb{X)}\}$$ for $\theta\in \Theta$ ? Where comes the expected value under the true $\theta$ from?
EDIT2:
$$P_\theta(\text{reject}) = E_\theta[1_{\text{reject}}] = E_\theta[E_\theta[1_{\text{reject}} \mid X]] = E_\theta[\psi(X)].$$
$$P_\theta(\text{reject}) = E_\theta[1_{\text{reject}}] = E_X[E_\theta[1_{\text{reject}} \mid X]] = E_X[\psi(X)].$$
$$P_\theta(\text{reject}) = E_\theta[1_{\text{reject}}] = E_\theta[E_X[1_{\text{reject}} \mid X]] = E_\theta[\psi(X)].$$
It depends on the definitions you are using, but really the best formulation is that we have some measure $\mathbb{P}$ on the underlying space of events where $E[1_A]:= \int 1_A d\mathbb{P} = \mathbb{P}(A)$. So it pops straight out of definitions if you define everything a certain way. If this doesn't make sense/or fit with your definitions, you'll have to give more context.