I am having troubles understanding the definition of a function being measurable. I have that for a measure space $(\Omega, \mathcal{F}, \mu)$ a function $f: \Omega \to \mathbb{R}$ is measurable if $\{x | f(x) > c\} \in \mathcal{F}$ for every $ c \in \mathbb{R}$
Now, I looked at an example which showed that if $f$ is an increasing function, then $f$ is measurable. It stated that since $f$ is increasing, the set $\{ x | f(x) > c\}$ is an interval, and since intervals are Lebesgue measurable we have that $f$ is Lebesgue measurable. I understand that $\{ x | f(x) > c\}$ is an interval, and intervals are Lebesgue measurable, but I don't see why this implies that $f$ is Lebesgue measurable. From the definition we require that $\{x | f(x) > c\} \in \mathcal{F}$ - What does this mean exactly? I thought it just meant that it's an element of the $\sigma$ algebra $\mathcal{F}$.
In the example you referred to, $\Omega$ is $\mathbb R$, $\mathcal F$ is the $\sigma$-algebra of Lebesgue measurable subsets of $\mathbb R$, and $\mu$ is Lebesgue measure on $\mathbb R$. So, the statement that $\{ x \mid f(x) > c \} \in \mathcal F$ means (in this example) that $\{ x \mid f(x) > c \}$ is a Lebesgue measurable subset of $\mathbb R$. If $f$ is increasing, then $\{ x \mid f(x) > c \}$ is an interval, and any interval is Lebesgue measurable.