Tail bound of sum less than sum of tail bounds

Question

Problem: Suppose we have a probability space $(\Omega, \mathcal{M}, P)$, a random variable $X$ on this space, and two nonnegative measurable functions $f,g:\mathbb{R}\to[0, \infty)$. Choose some $\epsilon > 0$. Prove that $$P(f(X)+g(X)>\epsilon) \le P(f(X)>\epsilon/2) + P(g(X)>\epsilon/2)$$

Attempt: by definition we know $P(f(X)+g(X)>\epsilon) = P(\{\nu \vert f(X)(\nu)+g(X)(\nu)))>\epsilon\})$, and it seems like a basic real analysis exercise, however I'm having trouble splitting this term into the two desired terms. Any help is appreciated!

Answer 1

Observe that $$ \{ \nu \in \Omega \mid f(X)(\nu) + g(X)(\nu) > \epsilon \} \subset \{ \nu \in \Omega \mid f(X) (\nu) > \tfrac \epsilon 2\} \ \cup \ \{ \nu \in \Omega \mid g(X) (\nu) > \tfrac \epsilon 2\}. $$

So \begin{align*} P\left( \{ \nu \in \Omega \mid f(X)(\nu) + g(X)(\nu) > \epsilon \}\right) & \leq P \left( \{ \nu \in \Omega \mid f(X) (\nu) > \tfrac \epsilon 2\} \ \cup \ \{ \nu \in \Omega \mid g(X) (\nu) > \tfrac \epsilon 2\} \right) \\ & \leq P \left( \{ \nu \in \Omega \mid f(X) (\nu) > \tfrac \epsilon 2\} \right) + P \left( \{ \nu \in \Omega \mid g(X) (\nu) > \tfrac \epsilon 2\} \right). \end{align*}