Dominated convergence question

[bbkrsen585]

Hi there. I was wondering if someone could help me out with the following.

Let E₁, E₂, ... be a sequence of nondecreasing measurable sets, each with finite measure.

Define E = \bigcupE_k, where E is the union of an infinite number of sets E_k.

Suppose f is measurable and Lebesgue integrable on each E_k.

Prove:

1) f is integrable on E if and only if \int_Ek|f| stays uniformly bounded (over all E_k).

2) Moreover, prove that if the equivalence in (1) holds then,

lim \int_Ek f = \int_E f.

 

[jbunniii]

It seems that the => direction of 1) is immediate from the DCT if you set

g_k = f \cdot 1_{E_k}

where

1_{E_k} is the characteristic function of E_k.

You can use f itself as the dominating function.

The other direction will take a bit more work.