Let f:[0,1] --> (0, ) be a (Lebesgue) integrable function

and define g to be: . Show that g is integrable.

Answer:

If g is a measurable function, I can easily show that it's integrable. It's my reasoning that g is measurable that may be suspect, I claim that:

Let O be an open set. We know that: is a measurable set.

Thus, , where = { }

So it remains to show that U measurable implies is measurable, which i was able to show but is a bit long, so I'll skip it.

Does this seem right, though? Or did i make it way too complicated?