Intuition behind Real Money Demand and Real Money Supply Standard undergraduate textbooks begin with an exposition of the  money market with the following equilibrium condititon: 

$$\frac{M^{s}}{P}=L\left(R,Y\right)
$$

In the above, the LHS is what is defined as "Real Money Supply", and the RHS is defined as "Real Money Demand". My question is the following: does the fact that we divide the LHS by the price level render these two quantities in units of goods? Intuitively, I think these quantities represent the number of goods that can be bought for a given amount of nominal money. What are the units of these quantities?
They're in monetary units! Let's say an economy has a money supply of $100$ $mu$ and a price level of $1$. Another economy has the same money supply but a price level of $1.5$. Which economy has the bigger real money supply? The first, obviously. In the first hypothetical economy, you can think of the average transaction as one involving smaller amounts of currency. Real money demand will of course depend on nominal income $Y$. An increase in nominal income makes people demand more money because there's more transactions to be made.