By Terence Tao
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Additional resources for An Introduction To Measure Theory (January 2011 Draft)
Intuitively, we want to build a set which has oscillatory structure even at arbitrarily fine scales. We will non-rigorously do this as follows. We will work inside the unit interval [0, 1]. For each x ∈ [0, 1], we imagine that we flip a coin to give either heads or tails (with an independent coin flip for each x), and let E ⊂ [0, 1] be the set of all the x ∈ [0, 1] for which the coin flip came up heads. We suppose for contradiction that E is Lebesgue measurable. g. 4]) in an extremely nonrigorous fashion, we thus expect m(E) to equal 1/2.
10. ) Show that the half-open interval [0, 1) cannot be expressed as the countable union of disjoint closed intervals. (Hint: It is easy to prevent [0, 1) from being expressed as the finite union of disjoint closed intervals. Next, assume for sake of contradiction that [0, 1) is the union of infinitely many closed intervals, and conclude that [0, 1) is homeomorphic to the middle thirds Cantor set, which is absurd. 7 of An epsilon of room, Vol. ) For an additional challenge, show that [0, 1) cannot be expressed as the countable union of disjoint closed sets.
Now we connect the Riemann integral to Jordan measure in two ways. 24 (Basic properties of the Riemann integral). Let [a, b] be an interval, and let f, g : [a, b] → R be Riemann integrable. Establish the following statements: (1) (Linearity) For any real number c, cf and f +g are Riemann b b b integrable, with a cf (x) dx = c · a f (x) dx and a f (x) + g(x) dx = b a f (x) dx + b a g(x) dx. e. f (x) ≤ g(x) for all b b x ∈ [a, b]) then a f (x) dx ≤ a g(x) dx. 8A function f : [a, b] → R is piecewise continuous if one can partition [a, b] into finitely many intervals, such that f is continuous on each interval.