# Risk-neutral measure(s) under collateralization and funding costs

In Piterbarg (2010) the author presents a modified Black-Scholes model with an economy with a CSA-collateral (OIS) rate $r_C(t)$, a repo rate $r_R(t)$ and considers a derivative $V(t)$ written on a stock share $S(t)$ and collateralized by $C(t)$. The market has two assets, a zero-coupon bond yielding the OIS rate $P_C(t)$ and the stock, both following under their respective risk-neutral measures a Geometric Brownian Motion with drifts the OIS rate and the repo rate respectively (assuming no dividends) $-$ numbering as in paper:

\begin{align} dP_C(t)/P_C(t) &= r_C(t)dt + \sigma_C(t)dW_C(t) \qquad \text{(1)} \\[6pt] dS(t)/S(t) &= r_R(t)dt+\sigma_S(t)dW_S(t) \qquad \text{(4)} \end{align}

The author defines one risk-neutral measure $P$ from the zero-coupon bond's Brownian Motion $W_C(t)$, however note that Piterbarg's hedging argument shows that the stock price is a martingale under the risk-neutral measure associated to the money market account with rate $r_R(t)$. Then the author states just below equation $\text{(4)}$:

Note that if our probability space [generated by the stock's Brownian Motion $W_S(t)$] is rich enough, we can take it to be the same risk-neutral measure $P$ as used in $\text{(1)}$.

What might the author mean by that sentence?