Lebesgue Outer Measure and Lebesgue Measurable Sets

Question

Question:

If m*(E) = 0, show that E is Lebesuge measurable, where m*(E) denote the Lebesgue Outer Measure of E.

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Notations I've used:

^ refers to intersection.

eg: A^E refers ro the intersection of the two sets A & E

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My approach towards the question:

We know that a subest E of R is Lebesuge measurable if,

m*(A) = m*(A^E) + m*(A^(E^c) --->(*) for any subset A of R.

Now consider m*(E).

m*(E) = 0 (Given) --->(1)

But,

m*(E^E) = m*(E) = 0 (given in the question)

and,

m*(E^(E^c)) =m*(null set) = 0

This gives,

m*(E^E) = m*(E) + m*(E^(E^c)) =0 --->(2)

From (1) & (2),

m*(E) = m*(E) + m*(E^(E^c)) which can be considered a special case of (*), which proves E is Lebesgue measurable when m*(E) = 0

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My concerns regarding my approach to this question are:

(1) Is this a correct way to prove this?

(2) Are there any other better ways to prove this?

Answer 1

As you have stated, the equation m*(A) = m*(A^E) + m*(A^(E^c)) has to hold for any subset A of R, but you only proved it for A = E. However, your approach is definitely on the right track.

In particular, to fill in the details for you, I'd argue the following.

I'm going to show two things:

(1) m*(A^E) = 0

(2) m*(A^(E^c)) = m*(A)

And these two will be enough to show that right-hand side of the equation above is equal to m*(A)

For (1) we note that m*(A^E) <= m^*(E) = 0 and since m*(A^E) >= 0 (since measures are always nonnegative), it follows that m*(A^E) = 0

For (2) we immediately note that m*(A^(E^c)) <= m*(A), and thus if we can show the reverse inequality we will be done. However, the reverse inequality follows from the following: m*(A^(E^c)) = m*(A\E) >= m*(A) - m*(E), and since m*(E) = 0, it follows that m*(A^(E^c)) >= m*(A) as desired.

As far as "other ways to prove it" I'm not aware of another method although the method you proposed seems to be standard (which is a good thing because it makes it applicable to a lot of other questions).

I know this is a lot so let me know if you have any follow-up questions!