Sandwich lemma for functions: Show that the integrals converge

Question

Let $g_j, f_j, h_j: R^n \to R$ be Lebesgue-integrable functions, $g_j \le f_j \le h_j $ nearly everywhere, and

$$ \lim_{j\to\infty} (g_j, f_j, h_j) = (g, f, h)$$

nearly everywhere. Also $h,g$ are integrable and

$$ \lim_{j\to\infty} \int_{R^n}g_j \, dL^n = \int_{R^j}g \, dL^n, \qquad \lim_{n\to\infty} \int_{R^n}h_j \, dL^n = \int_{R^n}h\, dL^n.$$

Show that

$$ \lim_{j\to\infty} \int_{R^n}f_j \, dL^n = \int_{R^n}f\, dL^n $$

Answer 1

Hint:

1. Apply Fatou's lemma to $u_j := h_j-f_j$ to conclude $$\int (h-f) \, dL^n \leq \liminf_{j \to \infty} \left( \int h_j \, dL^n - \int f_j \, dL^n \right).$$
2. Deduce from the first step that $$\int f \, dL^n \geq \limsup_{j \to \infty} \int f_j \, dL^n.$$
3. Apply the same argumentation to $v_j := f_j-g_j$ to conclude $$\int f \, dL^n \leq \liminf_{j \to \infty} \int f_j \, dL^n.$$