One-step Binomial model's Radon-Nikodym derivative
In economics, the Radon-Nikodym density $\frac{d \mathbb Q}{d \mathbb P}$ of the risk-neutral measure $\mathbb Q$ with respect to the physical measure $\mathbb P$ is the price of Arrow-Debreu securities. It is a price, not a claim. 

In the binomial setting, there are two AD securities, $1_u$ and $1_d$. The former entitles the holder to 1 unit of numeraire if, and only if, state $u$ realizes.
The no-arbitrage price of $1_u$ is $\frac{q_u}{p_u}1_u$ (say $R = 1$). Similarly for $1_d$.

In general, the price of the AD portfolio $1_{\Omega'}$---which pays off 1 unit of numeraire if, and only if, state $\omega \in \Omega'$ realizes---is
$$
E^{\mathbb P}[ 1_{\Omega'} \frac{d \mathbb Q}{d \mathbb P}] = \mathbb Q (\Omega').
$$
So 
$$
E^{\mathbb P}[\frac{d \mathbb Q}{d \mathbb P}] = 1
$$ 
is then the price of a bond/risk-free security (discount accordingly if $r \neq 1$).

Extending from $1_{\Omega'}$ to a general $X$ gives the usual risk-neutral pricing formula $E^{\mathbb Q}[X]$, as you have written out.