Monotone convergence theorem

If nonnegative measurable functions satisfy

f_{n} ↑ f

almost everywhere, then

\int f_{n} ↑ \int f

, allowing the value infinity.

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- Moving-spike obstruction to interchanging a limit and an infinite sum Monotone convergence theorem

Moving-spike obstruction to interchanging a limit and an infinite sum

Let

f (t, n) = 1_{[n, n + 1)} (t)

for

t \geq 1

. For every fixed

n

,

f (t, n) \to 0

t \to \infty

, but

\sum_{n} f (t, n) = 1

for every

t \geq 1

. Pointwise convergence without monotonicity or domination therefore does not justify interchanging a limit and an infinite sum.

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Monotone convergence theorem by

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The Monotone Convergence Theorem (MCT) is a fundamental result in measure theory and is especially important in the context of Lebesgue integration. The theorem provides conditions under which the limit of an increasing sequence of measurable functions converges to the integral of the limit function.

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