# Is there a way to measure the maximum random bits of the outputs of a generator?

I want to give examples to explain want I want to know first. Let $$G \colon s \mapsto G(s)$$ be a PRG.

1. Let $$F_{1} \colon s \mapsto G(s) \Vert b$$, where $$b = \bigoplus_{k = 1}^{|G(s)|} G(s)[k]$$. Obviously, $$F_{1}$$ is not a PRG, since the last bit can be predicted. But if we remove the last bit, it is a pseudorandom string.

2. Let $$F_{2} \colon s \mapsto G(s) \Vert G(s)$$, $$F_{2}$$ cannot be PRG, too. but if only choose $$|G(s)|$$ bits properly, it can be pseudorandom.

So, if $$F \colon \{0,1\}^l \to \{0,1\}^n$$ is a polytime deterministic generator. Perhaps, $$F$$ is not a PRG, but $$F$$ may be closed to a PRG. Namely, there exists a compression algorithm $$A \colon \{0,1\}^{n} \to \{0,1\}^m$$, where $$n \geq m > l$$, such that $$A \circ F$$ is a PRG. Is it well-defined? Can we use something like $$n - \min_{A} m$$ to measure how is $$F$$ closed to PRGs? Is there a standard way to measure the maximum random bits of all the outputs of $$F$$?