Perhaps other memeber of qSE are going to correct me, but I think the following rule of thumb is useful. Whenever you have a doubt, try to forget that a pricing measure is a probability measure. This is just a pricing tool: originally for any option/derivative/contingent claim we'd like to know its price, so we introduce a map $\pi:X\to \Bbb R$ such that $\pi(x)$ is the current price of the contingent claim $x$. For example, $x$ can be a call option with maturity of 1 year and ATM strike, or $x$ can be the futures contract expiring in 10 days. Now, it happens that $\pi$ is a linear functional on $X$, and $\pi(1) = 1$ - that is the value of the assets that will pay us $1$ under any circumstance is $1$ (let's assume discount rates are $0$). From that we see that $\pi$ is similar to an expectation operator, so we can define a corresponding probability measure - which we call a pricing (risk-neutral) measure. There are perhaps some deeper thought underlying such a coincidence, but for all philosophical questions: use pricing measure to find the price, to find the Greeks etc. For anything else use the physical measure.
Example: let's say we want to buy an option which we know we can't hedge perfectly, and estimate whether we can afford vacation on Hawaii after expiry. We do the following:
- Price the option (use pricing measure)
- Compute Greeks to hedge (use pricing measure)
- Run Monte-Carlo to estimate our losses/profits from imperfect hedge under the imperfect hedging strategy computed in step 2. (use real measure)