Let f be defined on $\mathbb{R}^d,$ suppose $\int_{\mathbb{R}^d} |f(x)|\,dx<\infty$. Is f then measurable?
My question boils down to "Do I have to check that a given function is measurable, or can I suppose the function is measurable, compute the integral, and if the integral is finite, observe that the function was indeed measurable?"