50 1. Measure theory |x| ≤ n} for some interval or ray Ic, and is thus measurable. As a conse- quence, fn is a simple function, and by construction it is bounded and has finite measure support. The claim follows. With these equivalent formulations, we can now generate plenty of mea- surable functions: Exercise 1.3.3. (i) Show that every continuous function f : Rd → [0, +∞] is measur- able. (ii) Show that every unsigned simple function is measurable. (iii) Show that the supremum, infimum, limit superior, or limit inferior of unsigned measurable functions is unsigned measurable. (iv) Show that an unsigned function that is equal almost everywhere to an unsigned measurable function, is itself measurable. (v) Show that if a sequence fn of unsigned measurable functions con- verges pointwise almost everywhere to an unsigned limit f, then f is also measurable. (vi) If f : Rd → [0, +∞] is measurable and φ: [0, +∞] → [0, +∞] is continuous, show that φ ◦ f : Rd → [0, +∞] is measurable. (vii) If f, g are unsigned measurable functions, show that f + g and fg are measurable. In view of Exercise 1.3.3(iv), one can define the concept of measurability for an unsigned function that is only defined almost everywhere on Rd, rather than everywhere on Rd, by extending that function arbitrarily to the null set where it is currently undefined. Exercise 1.3.4. Let f : Rd → [0, +∞]. Show that f is a bounded unsigned measurable function if and only if f is the uniform limit of bounded simple functions. Exercise 1.3.5. Show that an unsigned function f : Rd → [0, +∞] is a simple function if and only if it is measurable and takes on at most finitely many values. Exercise 1.3.6. Let f : Rd → [0, +∞] be an unsigned measurable function. Show that the region {(x, t) ∈ Rd × R : 0 ≤ t ≤ f(x)} is a measurable subset of Rd+1. (There is a converse to this statement, but we will wait until Exercise 1.7.24 to prove it, once we have the Fubini-Tonelli theorem (Corollary 1.7.23) available to us.)

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