Lebesgue integral on interval [a,b] Suppose $f$ and $g$ are non negative real functions defined on interval $[a,b]$.  $f$,$g$ $\in L^1([a,b])$
If there is a sequence of decreasing measureable sets $...\subset A_2\subset A_{1}$ ,such that $A_n \subset [a,b]$ for all n. 
On which $\int_{A_n}f d\lambda = \int_{A_n}g d\lambda $ 
Prove for $A = \bigcap_i A_i$,we have $\int_A f d\lambda = \int_A g d\lambda$.
Sorry for the elementary of the question, my attempt is using DCT,since $f\mathbb{1}_{A_n} \le f$ and $f \in L^1$.since $f\mathbb{1}_{A_n} \to f\mathbb{1}_A $ we have  $\int_A f d\lambda = \int_A g d\lambda$ right?

the measure $$\lim_{n \to \infty}\mu(A_n \bigcap A^c)=0$$
$$\lim_{n \to \infty}\int_A (f-g)d\lambda= \lim_{n \to \infty}(\int_{A_n} (f-g)d\lambda - \int_{A_n\bigcap A^c} (f-g)d\lambda)=0$$
