I've seen three different definitions for expected value of a random variable. The first one is the wikipedia's version: $E[X]=\int_\Omega X\,\mathrm{d}P\,$ (Lebesgue integral).
The second is of my first lecturer: $E[X]=-\int_{-\infty}^0 F_x(t) \,\mathrm{d}t+\int_0^\infty (1-F_x(t))\,\mathrm{d}t \hspace{2 mm}$ (using distribution function).
The third definition (used by my second lecturer) is this: $ E[X]=\int_{-\infty}^\infty \alpha\,\mathrm{d}F_x(\alpha)\,$ where the integral $ \int_{A}^B g(\alpha)\,\mathrm{d}F_x(\alpha)\,$ is defined as $ \int_{A}^B g(\alpha)\, \mathrm{d}F_x(\alpha) \hspace{2 mm} = \lim_{\Delta\alpha\rightarrow 0} \sum_i g(\alpha_i) (F(\alpha_{i+1})-F(\alpha_i)) \qquad \text{(Riemann integral)} . $
Who's right? and are all these three definitions equivalent?