Any nonnegative random variable $Z$ with expectation 1 is a Radon-Nikodym derivative: $$ \mathbb{E}^{\mathbb{P}} \left(Z\right) = \mathbb{E}^{\mathbb{P}} \left(\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\right) = \mathbb{E}^{\mathbb{Q}} \left(1\right) = \int{\mathrm{d}\mathbb{Q}} = 1 $$ $$ \mathbb{Q} \left(A\right) = \mathbb{E}^\mathbb{P} \left(Z 1_A\right) \in \left[0, 1\right] $$ If it is positive, the probability measure $\mathbb{Q}$ that $Z$ defines is equivalent to the original probability measure $\mathbb{P}$.

Now, by definition of a numeraire, under its associated probability measure, all asset prices expressed as units of the numeraire are martingales. For $\mathbb{Q}$ with numeraire $M$ and $N$ a positive asset price process, $$ \mathbb{E}^{\mathbb{Q}} \left(\frac{N_T}{M_T}\right) = \frac{N_0}{M_0} \Rightarrow \mathbb{E}^{\mathbb{Q}} \left(\frac{M_0}{M_T}\frac{N_T}{N_0}\right) = 1 $$ Since a numeraire is always chosen to be a strictly positive asset price process, the random variable $\frac{M_0}{M_T}\frac{N_T}{N_0}$ defines the Radon-Nikodym derivative of measure $\mathbb{Q}^N$ with respect to $\mathbb{Q}$. If $X$ is an (arbitrary) asset price process, $$ \mathbb{E}^{\mathbb{Q}^N} \left(X_T \frac{N_0}{N_T}\right) = \mathbb{E}^{\mathbb{Q}} \left(X_T \frac{N_0}{N_T}\frac{M_0}{M_T}\frac{N_T}{N_0}\right) = \mathbb{E}^{\mathbb{Q}} \left(X_T\frac{M_0}{M_T}\right) = X_0 $$ That shows that $N$ is indeed the numeraire for measure $\mathbb{Q}^N$.

Any nonnegative random variable $Z$ with expectation 1 is a Radon-Nikodym derivative: $$ \mathbb{E}^{\mathbb{P}} \left(Z\right) = \mathbb{E}^{\mathbb{P}} \left(\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}\right) = \mathbb{E}^{\mathbb{Q}} \left(1\right) = \int{\mathrm{d}\mathbb{Q}} = 1 $$ $$ \mathbb{Q} \left(A\right) = \mathbb{E}^\mathbb{P} \left(Z 1_A\right) \in \left[0, 1\right] $$ If $Z$ is positive, the probability measure $\mathbb{Q}$ that it defines is equivalent to the original probability measure $\mathbb{P}$.

Now, by definition of a numeraire, under its associated probability measure, all asset prices expressed as units of the numeraire are martingales. For $\mathbb{Q}$ with numeraire $M$ and $N$ a positive asset price process, $$ \mathbb{E}^{\mathbb{Q}} \left(\frac{N_T}{M_T}\right) = \frac{N_0}{M_0} \Rightarrow \mathbb{E}^{\mathbb{Q}} \left(\frac{M_0}{M_T}\frac{N_T}{N_0}\right) = 1 $$ Since a numeraire is always chosen to be a strictly positive asset price process, the random variable $\frac{M_0}{M_T}\frac{N_T}{N_0}$ defines the Radon-Nikodym derivative of measure $\mathbb{Q}^N$ with respect to $\mathbb{Q}$. If $X$ is an (arbitrary) asset price process, $$ \mathbb{E}^{\mathbb{Q}^N} \left(X_T \frac{N_0}{N_T}\right) = \mathbb{E}^{\mathbb{Q}} \left(X_T \frac{N_0}{N_T}\frac{M_0}{M_T}\frac{N_T}{N_0}\right) = \mathbb{E}^{\mathbb{Q}} \left(X_T\frac{M_0}{M_T}\right) = X_0 $$ That shows that $N$ is indeed the numeraire for measure $\mathbb{Q}^N$.