If $a < b$ and $b = \infty$, then $a < \infty$? A very simple question, but I am not sure for this moment.

I have a strict inequality $a &lt; b$. And I prove that $b = \infty$, say $b$ is an integral. Does this prove that $a &lt; \infty$, that is, $a$ is finite? Question seems simple but things can easily be messed up with infinity.

To be more clear, say $a = \int f dx$ and $b = \int g dx$ and I have $a &lt; b$. If $\int g dx = \infty$, can we say that $\int f dx &lt; \infty$?

If one can construct pathological counterexamples involve limits etc, I will be very happy.
Talking about $\int fdx$ makes no sense, as that is a concrete function, and not a number.

You might want to talk about the definite integral, $\int_0^\infty fdx$ and $\int_0^\infty gdx$. You may also want to discern between showing $\int_0^\infty fdx&lt;\int_0^\infty gdx$ (in which case your claim would be correct), and the case where $f(x)&lt;g(x)$ for all $x$ in the domain over which you are integrating. 

In the latter case take $f(x)=1$ and $g(x)=2$.
This assumption is only correct if $ (a,b) \in \mathbb{R}\times\mathbb{R} $. 

With $\infty$ in the game, this is no more correct. The concept of $\infty$ has some interesting properties among which is a lot of fluidness.

Actually: $\infty \lt \infty$, $\infty /\infty \ne 1$ and bunch of other crazy things are possible with $\infty$.