# Hypothesis space in AdaBoost or general Machine learning

I was curious about the following: in most learning algorithms, when an algorithm is said to learn a concept class $$C$$ then the algorithm outputs a function from the hypothesis space $$H$$ and often the complexity of the learning algorithm is stated in terms of the VC dimension of $$H$$. I was wondering, what is an assumption on $$H$$, for example, does it include all possible functions or does it only include $$C$$, is there any reference which talks about the properties of function in $$H$$ that I can assume? Say, for every $$c\in C$$, does there always exist an $$h\in H$$ for which $$Pr_{x} [h(x)=c(x)]\geq 1-\varepsilon$$ for every $$\varepsilon>0$$. Is this fair to assume?