In Moonlight Sonata there is a g sharp?

The G is not sharped in the key sig, but in one measure there is a G# and then another G in the same measure, but it doesn't say that is is sharped. Would I sharp it or no? Because in the next measure the first G is once again sharped and the second isn't. Do you have to specify a sharp on both notes for them both to be sharp or is it different because they're in the same measure?
 
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