Actually calculating the Lebesgue Outer Measure of a set

[jdinatale]

If you are unfamiliar with that notation, S(A, B) = (A \ B) U (B \ A), which is the symmetric difference.

And D(A, B) = m^*(S(A, B)), which is the outer measure of the symmetric difference.

My issue in this calculation is this: outer measure of a set A is defined in terms of a countable covering of A with intervals I_k. How do you cover something in R^2 with intervals?

My guess on how to solve the problem: D(A, B) = m^*(S(A, B)) = m^*((0, 1]X[0, 1] U [1, 4]X(1, 10) U [0, 1]X[2, 10]) = m^*(0, 1]X[0, 1] + m^*[1, 4]X(1, 10) + m^*[0, 1]X[2, 10] = 1 + 27 + 8 = 36

 

[jgens]

My issue in this calculation is this: outer measure of a set A is defined in terms of a countable covering of A with intervals I_k. How do you cover something in R^2 with intervals?

The outer Lebesgue measure on $\mathbb{R}$ can be defined in terms of a countable covering by intervals. The outer Lebesgue measure on $\mathbb{R}^2$ can be defined in terms of a countable covering by rectangles. This should clear up your confusion.

If you want more information about this, you can read the following wikipedia article: http://en.wikipedia.org/wiki/Lebesgue_measure#Construction_of_the_Lebesgue_measure