Suppose that the function f is defined on a measurable set E and has the property that {x ∈ E|f (x) > c} is measurable for each rational number c. Is f necessarily measurable? Prove your result. Let f be a function with measurable domain D. Show that f is measurable if and only if the function g defined on R by g(x) = f (x) for x ∈ D and g(x) = 0 for x /∈ D is measurable.

Suppose that the function f is defined on a measurable set E and has the property that {x ∈
E|f (x) > c} is measurable for each rational number c. Is f necessarily measurable? Prove your
result.

Let f be a function with measurable domain D. Show that f is measurable if and only if the
function g defined on R by g(x) = f (x) for x ∈ D and g(x) = 0 for x /∈ D is measurable.