In Kerry Back, *A Course in Derivative Securities*, Sect. 1.4 (page 29), the author stated the FTAP in the following form (in boldface):

> 
> If there are no arbitrage opportunities,
>  then for each (non-dividend-paying) asset, there exists a probability
>  measure such that the ratio of any other (non-dividend-paying)
>  asset price to the first (numeraire) asset price is a martingale.
> 
> 
> 

And he commented that

> 
> We have applied this statement to the risk-free asset, which pays dividends (interest).
>  However, the price $R\_u = R\_d = e^{rT}$ includes the interest, so no interest has
>  been withdrawn—the interest has been reinvested—prior to the maturity T of the
>  option. This is what we mean by a “non-dividend-paying” asset. In general, we will
>  apply the formulas developed in this and the following section to dividend-paying
>  assets by considering the portfolios in which dividends are reinvested.
> 
> 
> 

In such spirits, when we want to use a dividend paying stock $S\_t$ as numeraire we should actually use the "reinvested" asset $e^{qt}S\_t$ as the numeraire instead.

My question is, what would be the consequences of really using a div-paying asset, such as $S\_t$ (when $q>0$) as numeraire? Is it possible to make the corresponding martingale measure not exist, for example? (But in my opinion, the existence of a martingale measure seems only to rely on positivity...)