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QUIZ 4. MEASURE AND INTEGRATION Let (X, C, µ) be a fixed measure space. (a) Let E and F be C−measurable sets, E ⊂ F . Show that µ(E) ≤ µ(F ). Also show that µ(F \ E) = µ(F ) − µ(E) if µ(E) < +∞. (b) What does it mean that a function f is equal to 0 µ−almost everywhere? (c) What does it mean that a simple function φ is written on standard form? (d)How is the integral of a simple measurable (nonnegative) function φ defined, and how can we define the integral of a non-negative measurable function f ? 1