Dunford and Schwartz. First, we may assume that . Reason: let (with if ). In other words, is the complex number of absolute value 1 such that . Replace by , which has the same norm as in any of the spaces, and is always real and non-negative. We can then assume thatfor all non-negative . (We are only told that this holds for bounded , but you can easily see from the Monotone Convergence theorem that it holds for all .)
Now comes the clever part. It follows by putting in (*) that . Also, , with . Therefore, putting in (*), we get .
Now let . We have shown that , with . So , and we can repeat the argument in the previous paragraph to get . Thus , with .
Continue in that way to see that for n=1,2,3,... . But . By Fatou's lemma, we can let and get , from which with .