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Let epsilon 0 since mu is already a radon measure, for any E, we can find U open and E subset U such that mu(U)-mu(E), then nu(U)-nu(E) By corollary 3.6, but since v(U) v(E) for all such U.

If anyone can see what went wrong here I would appreciate it if you can let me know.

Note that En is finite(and hence, sigma -finite) because phi is in L1.

Since varepsilon 0 is arbitrary and N only depends on varepsilon, we have proven the inner regularity on the Borel set.

Then, we can find a compact set K such that K subseteq Ec and nu(Ec - K).

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Folland Solutions Chapter 7