hw3_3.pdf

# hw3_3.pdf - Austin Mohr Math 704 Homework 3 Problem 1 Show...

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Austin Mohr Math 704 Homework 3 Problem 1 Show that there exist closed sets A and B with m ( A ) = m ( B ) = 0, but m ( A + B ) > 0: a. In R , let A = C (the Cantor set), B = C 2 . Proof. Recall that the Cantor set (here, A ) contains no open interval, and so has measure 0. Similarly, B has measure 0. Recall also that an element belongs to A if and only if it has a ternary expansion using only 0s and 2s. Hence, an element belongs to B if and only if it has a ternary expansion using only 0s and 1s. Now, let x [0 , 1]. We choose a A and b B such that x = a + b (and so show that x A + B ) as follows: If the k th digit of x is 0, specify that the k th digit of a is 0 and the k th digit of b is 0. If the k th digit of x is 1, specify that the k th digit of a is 0 and the k th digit of b is 1. If the k th digit of x is 2, specify that the k th digit of a is 2 and the k th digit of b is 0. Hence, A + B [0 , 1], and so by problem 4a, we have that A + B is measurable (since A + B contains a subset of nonzero measure). Furthermore, m ( A + B ) m ([0 , 1]) = 1.
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