IGNOU MMT 4 2026 - Verify the hypothesis and conclusions of the Fatou’s lemma

Question

Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence $\{f_n\}$ given by

$$f_n(x) = 2n \text{ for } x \in \left( \frac{1}{2n}, \frac{1}{n} \right)$

$$= 0 \text{ for } x \in \left( 0, \frac{1}{2n} \right] \cup \left[ \frac{1}{n}, 1 \right]$

09 Jan 2026

Answer :

Word Count : 296

We are asked to verify Fatou’s lemma for the sequence of functions [ f_n(x) = \begin{cases} 2n, & x \in \left(\frac{1}{2n}, \frac{1}{n}\right) [2mm] 0, & x \in \left(0, \frac{1}{2n}\right] \cup \left[\frac{1}{n}, 1\right] \end{cases} ] on the interval ([0,1]) with the Lebesgue measure. Let's go step by step. --- Step 1: Compute the integral of (f_n). [ \int_0^1 f_n(x),dx = \int_{\frac{1}{2n}}^{\frac{1}{n}} 2n , dx = 2n \cdot \left(\frac{1}{n} -

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