What is the equation $\mathbb{E}[mR]=1$? I have just joined this Stack Exchange site and I was reading the following question: What are the foundational equations of Economics?. I have seen a couple of times the following equation in that post:

$$\mathbb{E}[mR]=1$$

I have a certain idea of what it might mean, but could someone explain its meaning (or at least what do $m$ and $R$ stand for? It is difficult to Google it if I don't know what these 2 quantities are).

Also, is the expectation $\mathbb{E}[\cdot]$ taken w.r.t to any particular probability measure, or simply with respect to the real world measure?

PS: I apologize if the tags are not well-suited to the question, I have tried to guess what this equation might be related to.
This is an important result in financial economics (asset pricing) but not trivial to explain intuitively. I do my best to give you the big picture and get you started on your research. 

R is the return on an asset or portfolio.  Any asset or portfolio.  

m is the stochastic discount factor or pricing kernel. 

The expectation here is over possible states of the world given information at time t (m and R are as of t+1).  

These are physical, not risk-neutral, probabilities. In fact, a (positive) stochastic discount factor is what is used to translate physical to risk-neutral probabilities.  In a discrete case, the risk-neutral probability is the risk free rate times the physical probability times the value of the stochastic discount factor in that state.

The stochastic discount factor takes on a different value for each possible state of the world (that's the sense in which it is stochastic).  But it doesn't change for each different asset.  If it exists, a stochastic discount factor can be applied to every asset in the economy and price them all correctly.  

Facts about the SDF:


The existence of such a factor is equivalent to the law of one price holding.  
If it is always positive, there are no arbitrage opportunities in financial markets.  
If and only if markets are complete, then the stochastic discount factor is unique.
The expectation of the SDF is the risk-free rate


There are a lot of theoretical uses of this relation and important related results---many fundamental results in finance can be interpreted in an SDF context, though many of those results came before development of SDF theory.  For many common utility functions we can derive the form of the SDF for a representative agent theoretically.

I'm sorry if my answer isn't super accessible.  The equation is simple, but stochastic discount factor theory can be a little opaque.

Resources to consider reading:

Cochrane and Culp's chapter in Modern Risk Management: A History

Asset Pricing and Portfolio Choice Theory, by Kerry Back