Caratheodory Extension Theorem tells us that we can "extend" our measure on algebra to measure on $\sigma$-algebra. So what we extend in this case is the set on which we can use this measure (measure is the same). Conclusion: the Theorem is called "extension" because we extend algebra to $\sigma$-algebra (generated by this algebra). Am I right?
I usually see the words "extension" and "restriction". When we try to construct a Lebesgue Measure from outer measure we "restrict" outer measure to the $\sigma$-algebra. So simultaneously we "restrict" outer measure to $\sigma$-algebra and "extend" algebra to $\sigma$-algebra?