Determine if an infinite measurable set must contain a closed subset of infinite measure | Step-by-Step Solution

What We're Solving

We need to investigate whether every measurable set with infinite Lebesgue measure must contain a closed subset that also has infinite measure. This is a fascinating question that tests our understanding of the relationship between different types of sets in measure theory!

The Approach

This is a counterexample problem - we're going to show that the statement is FALSE by constructing a specific measurable set with infinite measure that contains NO closed subset with infinite measure. The key insight is to think about what makes closed sets "well-behaved" versus what allows measurable sets to be more "wild."

Step-by-Step Solution

Step 1: Understanding what we need to construct We need a measurable set E with:

• m(E) = ∞ (infinite Lebesgue measure)
• Every closed subset F ⊆ E has m(F) < ∞ (finite measure)

Step 2: The key insight - think "sparse but everywhere" The secret is to create a set that's infinite in measure but "spread out" in such a way that any closed subset can't capture too much of it. Think about how closed sets are "chunky" (they contain neighborhoods), while we want our set to be more "scattered."

Step 3: Constructing our counterexample Let's define: E = ⋃(n=1 to ∞) [n, n + 1 - 1/n)

This is the union of intervals:

• [1, 2) (length = 1)
• [2, 2.5) (length = 1/2)
• [3, 8/3) (length = 2/3)
• [4, 15/4) (length = 3/4)
• And so on...

Step 4: Verify E has infinite measure m(E) = ∑(n=1 to ∞) (1 - 1/n) = ∑(n=1 to ∞) (n-1)/n

Since this sum diverges (it behaves like the harmonic series), m(E) = ∞. ✓

Step 5: Show every closed subset has finite measure Here's the crucial part! Let F ⊆ E be any closed subset.

For each interval [n, n + 1 - 1/n), notice that:

• The right endpoint n + 1 - 1/n is NOT in our set E
• Since F is closed and F ⊆ E, F cannot contain this endpoint
• Therefore, F ∩ [n, n + 1 - 1/n) must be bounded away from the right endpoint

This means for each n, the intersection F ∩ [n, n + 1 - 1/n) has measure strictly less than 1 - 1/n.

Since F is closed, only finitely many of these intersections can have "substantial" measure, so m(F) < ∞.

The Answer

NO - not every measurable set with infinite measure contains a closed subset with infinite measure. Our counterexample E = ⋃(n=1 to ∞) [n, n + 1 - 1/n) demonstrates this beautifully!

Memory Tip

Remember this concept with "Sparse Infinity" - you can have infinite measure by spreading it out over infinitely many "almost complete" intervals, but closed sets can't grab enough of this scattered measure to become infinite themselves. It's like trying to catch an infinite amount of water with a finite net - the water keeps slipping through the gaps!

This problem shows how measure theory can be surprisingly subtle - intuition from finite settings doesn't always carry over to the infinite case! 🎓