Calculate the measure of a set

Question

I try to understand the following proof:

Now I don't understand the inequality in red. How can we conclude that or more precisely how can we calculate $\mu(A_k)$? Many thanks for some help.

Answer 1

Fix $k\in\mathbb N$. Clearly $X$ can be partitioned into $A_k$ and it's complement. Thus $$\int_X\vert f_{n_k}(x) - f_{n_{k+1}}(x)\vert^p\,\mathrm d\mu = \int_{A_k}\vert f_{n_k}(x) - f_{n_{k+1}}(x)\vert^p\,\mathrm d\mu + \int_{X\setminus A_k}\vert f_{n_k}(x) - f_{n_{k+1}}(x)\vert^p\,\mathrm d\mu$$ The second term is clearly larger than 0, so we can ignore it. The integrand in the first term is larger than $\frac{1}{2^k}$, so $$\int_{A_k}\vert f_{n_k}(x) - f_{n_{k+1}}(x)\vert^p\,\mathrm d\mu\geq\int_{A_k}\frac{1}{2^k}\,\mathrm d\mu = \frac{1}{2^k}\mu(A_k)$$