Why can the equal sign $(=)$ be included in the below inequality?

Question

Let $\left(X_n\right)_{n\geq1}$ be a sequence of random variables defined on a given fixed probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$.
Consider that $\mathbb{P}\left(\vert X_n-X\vert>\varepsilon\right)=\mathbb{E}\{1_{\{\vert X_n-X\vert >\varepsilon\}}\}$. Then, consider the event $\{\omega:\vert X_n-X\vert>\varepsilon\}$. One is allowed to state that $$\{\omega:\vert X_n-X\vert>\varepsilon\}\Leftrightarrow\Bigg\{\omega:\frac{\vert X_n-X\vert^p}{\varepsilon^p}>1\Bigg\}$$ Henceforth $\dfrac{\vert X_n-X\vert^p}{\varepsilon^p}>1$ on the event $\{\omega:\vert X_n-X\vert>\varepsilon\}= \{\vert X_n-X\vert>\varepsilon\}$. So, I think I could state that the below inequality holds true $$\mathbb{P}\left(\vert X_n-X\vert>\varepsilon\right)=\mathbb{E}\{1_{\{\vert X_n-X\vert >\varepsilon\}}\}<\mathbb{E}\Bigg\{\frac{\vert X_n-X\vert^p}{\varepsilon^p}1_{\{\vert X_n-X\vert >\varepsilon\}}\Bigg\}$$ However, on my book I read $$\mathbb{P}\left(\vert X_n-X\vert>\varepsilon\right)=\mathbb{E}\{1_{\{\vert X_n-X\vert >\varepsilon\}}\}\leq\mathbb{E}\Bigg\{\frac{\vert X_n-X\vert^p}{\varepsilon^p}1_{\{\vert X_n-X\vert >\varepsilon\}}\Bigg\}$$ So, my question is: why is equal sign $(=)$ included in the inequality on my book? I cannot figure out why one could be allowed to include it, given the above reasoning.

Answer 1

Your question seems to be when does the expectation preserve strict inequality. Here is a sufficient condition: When does the integral preserve strict inequalities? This might be useful for future reference.

As you can see, this condition is not satisfied here. One of the possible problems is that $$\mathbb{P}(|X_n-X| > \epsilon) = 0$$ is possible (for example, when $X_n = X$ a.s.) and then both sides are $0$, so we have equality. Thus strict inequality cannot hold in general

Answer 2

I was surprised how often I had this question when teaching analysis.

The notation $x < y$ means ''$x$ is less than $y$''.

The notation $x \leq y$ means ''$x$ is less than or equal to $y$'',

i.e. $x$ is less than $y$ or $x$ is equal to $y$.

Answer 3

The strict inequality doesn't hold when $\mathbb{P}(|X_n - X|>\varepsilon) = 0$ because the integral of a strictly positive function on a set of measure $0$ is $0$.