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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?

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Yes, although you should not say that the "measure is the same" (in fact in general the extension is not unique, but it is defined elsewhere anyway).

Also yes for your second question, although again a minor glitch: "the" $\sigma$-algebra is not unique.

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