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## Homework Statement

Let [tex](X,\Sigma,\mu)[/tex] be a measure space. Suppose that {

*f*

_{n}} is a sequence of nonnegative measurable functions, {

*f*

_{n}} converges to

*f*pointwise, and [tex] \int_X f = \lim\int_X f_n < \infty[/tex]. Prove that [tex] \int_E f = \lim\int_E f_n[/tex] for all [tex]E\in\Sigma[/tex]. Show by example that this need not be true if [tex] \int_X f = \lim\int_X f_n = \infty[/tex].

## Homework Equations

## The Attempt at a Solution

Since [tex]\lim\int_X f_n < \infty[/tex], I think that the sequence [tex]\{\int_X f_n\}[/tex] is bounded, say by

*M*. If I define

*g*(

*x*) to be the constant function

*g*(

*x*)=

*M*, can I just apply Lebesgue's Dominated Convergence Theorem?