Let $(X,M,mu)$ be a measure space and let $f:X to R$ be a measurable function.Prove that the positive measure $mu$ is finite

Let $(X,M,\mu)$ be a measure space and let $f:X \to \mathbb{R}$ be a measurable function. Assume that $f\in L^{1}(\mu)$ and $f-1 \in L^{p}(\mu)$ for some number $p\in [1,\infty)$. Prove that the positive measure $\mu$ is finite, that is $\mu(X) < \infty$.

Consider sets $\{x \in X: f(x) \geq 1/2\}$ and $\{x \in X: f(x)<1/2\}$.

Can someone pls help me with this problem, I am completely lost here