Yes, stabilizer protects chlorine from the sun. It is also known as cyanuric acid or CYA for short. You want to have 50 ppm of CYA for your sunny location. PH at 7.8 is fine. More here about how to add CYA to your pool, Pool School - Recommended Pool Chemicals
Without any stabilizer in the pool water, the half-life of chlorine is about 35mins give or take. So, in a little over four hours of direct noon time sun, you will lose 94% of the chlorine in your pool. SO the answer to your question is - yes, stabilizer is very important.