2. (a) Give the definition of a Lebesgue measurable function defined on Rd. (b) Prove that...

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2. (a) Give the definition of a Lebesgue measurable function defined on Rd. (b) Prove that if f and g are measurable functions, then: (i) The integer powers fk, k ≥ 1, are measurable. (ii) The functions f + g and fg are measurable if both f and g are finite-valued. (c) Let X[0,1] be the characteristic function of [0, 1]. Show that there is no everywhere continuous function f on R such that f(x)=x[0,1] (x) almost everywhere. (d) Given any measurable set E and any e > 0, show that there exists a step function of the form = 1 XR,, where the R., are rectangles, such that m({2:XE(c) # (c)}) < e. 2. (a) Give the definition of a Lebesgue measurable function defined on Rd. (b) Prove that if f and g are measurable functions, then: (i) The integer powers fk, k ≥ 1, are measurable. (ii) The functions f + g and fg are measurable if both f and g are finite-valued. (c) Let X[0,1] be the characteristic function of [0, 1]. Show that there is no everywhere continuous function f on R such that f(x)=x[0,1] (x) almost everywhere. (d) Given any measurable set E and any e > 0, show that there exists a step function of the form = 1 XR,, where the R., are rectangles, such that m({2:XE(c) # (c)}) < e.

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a A Lebesgue measurable function defined on Rd is a function for which preimages of open sets closed ... View the full answer