Let $(\Omega,\mathcal{F},P)$ be a probability space, $X:\Omega\to\mathbb{R^+}$ be a non-negative $\mathcal{F}-$measurable random variable with $E[X]=\infty$ and $\mathcal{H}\subset\mathcal{F}$ be a sub-$\sigma-$algebra. How can we show the existence of $E[X|\mathcal{H}]$?
For a real valued random variable with finite expectation I can understand the proof from wikipedia. It defines a measure $\mu^X:F\in\mathcal{F}\mapsto\int_F XdP$ and restricts it on $\mathcal{H}$. Then use the Radon–Nikodym derivative to show the existence. In case $E[X]=\infty$ the measure $\mu^X$ is not finite anymore. However, if we impose it to be $\sigma-$finite we can still use the Radon–Nikodym derivative.
I do not think it is true in general that such $\mu^X$ is $\sigma-$finite. Is there a way to show existence for such a random variable without this assumption?