Question

Suppose (X, S, μ) is a measure space and f1, f2, . . . is a

sequence of nonnegative S-measurable functions. Define f : X → [0,
[infinity]] by f (x) = ∑[infinity]k=1 f_k(x). Prove that I know I can use
the MCT

Answer 1

To prove that the function
`f(x) = ∑k=1 f_k(x)`n use the Monotone Convergence Theorem (MCT). The Monotone Convergence Theorem states that if we have a sequence of nonnegative measurable functions {f_n} and
`f(x) = ∑n=1 f_n(x)`able.

Step-by-step explanation:

To prove that the function
`f(x) = ∑k=1 f_k(x)` can use the Monotone Convergence Theorem (MCT).

The Monotone Convergence Theorem states that if we have a sequence of nonnegative measurable functions {fn} and
`f(x) = ∑n=1 f_n(x)`able.

The proof involves showing that for any positive integer m, the set x is measurable. This can be done by showing that x is measurable for each n, and then taking the union over all n.